unit 5
the first derivative test is when we use the first derivative to...
"test" whether or not a function has a maximum or minimum
6. given the function g(x) = -x^4 + 2x^2 - 1, find the interval(s) where g is concave up and increasing at the same time
(0,√1/3)
at what values of x does f(x) = x^4 - 8x^2 have a relative minimum
-2 and 2 only
what is the maximum value of the derivative of f(x) = 3x^2 - x^3
3
a rectangle is formed with the base on the x-axis and the top corners on the function y = 36 - x^2. what length and width should the rectangle have so that its area is a maximum?
A = 4√3 X 24
find the max area of a rectangle that can fit between the x-axis and the graph of y = 25-x^2
A = 96.225 un^2
a local wild board population is changing at a rate modeled by b(t) = 0.04t^4 - 0.25t^2 - 0.02t boar per year where t is measured in years. is the board population growing or shrinking at time t = 3 years? justify your answer.
b(t) is the rate of change, therefore the population is increasing because b(3)=0.93>0
candidates for absolute extrema (on an interval)
critical points endpoints
10. consider the function f(x) = {x^2, 0≤x<1; 0, 1≤x≤2. which of the following is true?
f attains an absolute minimum value of 0
if f is concave up, then f' is... if f is concave down, then f' is...
increasing decreasing
what is concavity?
the state or quality of being concave
18. write an equation of the Lin tangent to y = x^3 - 3x^2 - 4 at its point of inflection.
y + 6 = -3(x - 1)
1. find all the critical points for f(x) = 2/3x^3 + 5x^2 - 28x - 10. 2. find an inflection point(s) for f(x) = 2x(x+4)^3 3. determine the interval on which f(x) = x^3 - x^2 increases and decreases 4. if f'(x) = 2x^2 - 5 find the interval(s) where f is decreasing. 5. if f(x) = Ix^2 - 4I which of the following is true?
1. x = -2, x = -7 2. x = -4, x= -2 3. decreasing: (0,2/3) b/c f'(x)<0 increasing: (-∞,0) and (2/3,∞) b/c f'(x)>0 4. f(x) is decreasing on (-√5/2,√5/2) b/c f'(x)<0 5. f has points of inflection at +/-2; f has a relative maximum at (0,4)
does the line tangent to the graph of h at the given value of x lie above or below the graph of h? why? 10. h(x) = (1-x^2)^2 at x = -3 11. h(x) = -cos(2x) at x = π/3
10. below b/c h"(-3)>0 11. above b/c h"(π/3)<0
find the slope of the tangent line to (4-x)y^2 = x^3 at the point (2,2)
2
look at #3 and #4 not he front page of the mid-unit 5 CA
YUH
go to the back page of 5.1 and 5.2 review to do problems 1 and 2 with graphs.
YUH!!!!!
critical point
a point that has a possibility of being an extrema (max or min)
if the position of a particle is given by x(t) = t^3 - 12t^2 = -36t + 18, where t>0, a. find the point at which the particle changes direction. b. find the period of time during which the particle is slowing down.
a. t = 2, t = 6 b. slowing down (0,2) and (4,6) b/c x'(t) and x"(t) have different signs
f'(c) =
f(b)-f(a) / b-a
2. a particle's position along the y-axis is measured by y(t) = t - 3(t - 4)^(1/3), where t>0. find the intervals where the particle is speeding up. find the intervals where the particle is slowing down.
speeding up: (3,4) and (5,∞) slowing down: (0,3) and (4,5)
look at #3 on top 5.1 - mean value theorem - calc medic for the graph problem
YUH!!!!!!
look at 16 on page 4 of 5.4 for the graph
YUH!!!!!!
11. x: 1, 2, 3, 4 f(x): 3, 6, 5, -2 f'(x): 8, 3, -3, 6 g(x): 2, 1, 6, 3 g'(x): 4, 2, 3, 5 the functions f and g are differentiable for all real numbers. the table above gives values of the functions and their first derivatives at selected values of x. the function h is given by h(x) = f(g(x)) + 2. must there be a value c for 2<c<4 such that h'(c) = 1
yes, the MVT proves it
look at the beginning of 5.7 in the packet for the graph
yuh
look on the back of 5.6 determining concavity sheet to do graph stuff
yuh
look in the second part of the review packet for two graphs and a chart!!!!!!
yuh!!!!!!!!
1. for the function f(x) = 2x^3 + 4x^2 + 2x, use a table to help you organize and draw conclusions when is f both concave up and decreasing?
(-2/3,-1/3)
9. the table below gives selected values of a function f. the function is twice differentiable with f''(x)>0. x: 3, 5, 7 f(x): 12.5, 13.9, 16.1 which of the following could be the value of f'(5)?
0.9
on the graph of f(x), on a max, f'(x) = and f"(x) and on a min, f'(x) = and f"(x)
0; <0 0; >0
1. consider the curve 3x^3 + 3 = ln(4y^2) in the xy-plane. at the point (-1,1/2), is the curve increasing or decreasing? 2. consider the curve x^2 - 3 = e^y in the xy-plane. at the point (-2,0), is the curve concave up or down? 3. consider the curve y^3 - y = x^2 in the xy-plane. it is known that dy/dx = 2x/(3y^2 - 1) and d^2y/dx^2 = (2/(3y^2 - 1)) - (24x^2y)/((3y^2 - 1)^2). at the point (0,1) on the curve, is the point a relative max, relative min, or neither? justify.
1. increasing because dy/dx>0 2. concave down because d^2y/dx^2<0 3. min at (0,1) b/c dy/dx = 0 and d^2y/dx^2>0
find the points of inflection for each function. justify your answer 1. f(x) = 5 + 3x^2 - x^3 2. h(x) = (3x^2 - 2)^2 3. f(x) = x + 2sinx on the interval (0,2π)
1. point of inflection at x = 1 b/c f'' changes sign 2. points of inflection at x = -0.471 and x = 0.471 b/c h" changes sign 3. points of inflection at x = 0, x = π, and x = 2π b/c f" changes sign
for each table, selected values of x and f(x) are given. assume that f'(x) and f"(x) do not change signs. 3. x: 4, 5, 6, 7 f(x): -5, -8, -12, -17 a. is f(x) increasing or decreasing? b. is f(x) concave up or concave down? 4. x: -3, -2, -1, 0 f(x): -2, 3, 7, 10 a. is f(x) increasing or decreasing? b. is f(x) concave up or concave down? 5. x: 2, 3, 4, 5 f(x): 3, 0, -2, -3 a. is f(x) increasing or decreasing? b. is f(x) concave up or concave down?
3. decreasing; concave down 4. increasing; concave down 5. decreasing; concave up
1. let f(x) be a twice differentiable function such that f(4) = 7 and f(7) = 4. show that there must be a value c for 4<c<7, such that f'(c) = -1.
4-7 / 7-4 = -3/3 = -1
state the intervals of concavity 4. x(t) = t^3 - 15t^2 5. g(x) = cos(x/2) on the interval (0,2π)
4. concave down on (-∞,5) b/c x"<0; concave up on (5,∞) b/c x">0 5. concave down on (-∞,π) and (3π,∞) b/c g"<0; concave up on (π,3π) b/c g">0
the derivative f' is given for each problem. use a calculator to help you answer each question about f 7. f'(x) = (x + 3e^(-x))/(x^2 + 0.8). on what intervals is f increasing? 8. f'(x) = -sin - xcosx for 0≤x≤π. on which interval(s) is f decreasing? 9. f'(x) = (1/x) - e^(x)sinx for 0<x≤4. on what intervals is f decreasing?
7. (-∞,∞) 8. (0, 2.029) 9. (0.727, 3.128)
7. consider the curve defined by x^2 - y^2 - 5xy = 25 a. show that dy/dx = (2x-5y)/(5x+2y) b. find the slope of the line tangent to curve at each point on the curve when x = 2 c. find the positive value of x at which the curve has a vertical tangent line. show the work that leads to your answer. d. let x and y be functions of time t that are related by the equation x^2 - y^2 - 5xy = 25. at time t = 3, the value of x is 5, the value of y is 0, and the value of dy/dx is -2. find the value of dx/dt at time t = 3
7. a. done b. 39/-4 c. x = √(100/29) d. dx/dt=-5
f(x) is increasing if f'(x) f(x) is decreasing if f'(x)
>0 <0
16. let f"(x) = sinx^2. which of the following statements are true? I. f is concave up on (0, 1.77) and (2.51, 3.06) II. f is concave down on (1.78, 2.50) III. f' is increasing on (0, 1.77)
I, II, and III
imagine that you have at 28cm X 20cm sheet of metal. you cut out square corners and fold the corners up. the square cut-outs are not used and are recycled for other projects. you want to maximize the volume of the box you make. find the max volume.
V = 961.31cm^3
1. find the absolute maximum value and the absolute minimum value of the function f(x) = x^3 - 3x^2 + 1 on the interval [-1/2, 4]. remember to show that you checked all the candidates.
absolute max value is 17 absolute min value is -3
17. consider the differential equation dy/dx = 4x + y. find d^2y/dx^2. determine the concavity of all solution curves for the given differential equation in Quadrant I. give a reason for you answer.
concave up because d^2y/dx^2 > 0 if x and y are positive
implicit relationships still follow the same rules as functions. if dy/dx = 0 or dy/dx does not exist at a point, then that point is a... if d^2y/dx^2>0, then the graph is...
critical point concave up at that point
in the xy-plane, the graph of the twice differentiable function y = f(x) is concave down on the open interval (1,3) and is tangent to the line y = 4x+3 at x = 2. which of the following statements must be true about the derivative of f?
f'(x) ≤ 4 on the interval (2,2.1)
the mean value theorem can be applied to which of the following function on the closed interval [0,5]?
f(x) = (x-3)/(x+3)
how to do the second derivative test
find first derivative find zeros of first derivative find second derivative plug in the zeros into the second derivative find relative min/max with answers^ (positive or negative) justify using f'(#) = 0 and f"(#)<>0
mean value theorem
if a function f is continuous over the interval [a,b] (close interval) and differentiable over the interval (Abraham) then there exists a point within that open interval where the instantaneous rate of change equals the average rate of change over the interval.
3. for the given function f(x), f'(x) and f"(x) do not change signs. a table of values for f(x) is given in the table below. x: 0, 1, 2, 3 f(x): 1, 2, 4, 7 a. is f(x) increasing or decreasing b. is f(x) concave up or concave down
increasing concave up (because f(x) is increasing at an increasing rate)
when the slope of a function is positive, the function is... when the slope of a function is negative, the function is...
increasing decreasing
the rate of money in a particular mutual fund is represented by m(t) = sin(e/3)^t thousand dollars per year where t is measured in years. is the amount of money from this mutual fund increasing or decreasing at time t = 4 years? justify your answer.
increasing because the rate m(4)=0.384>0
if a function is concave up (like a parabola), what is f' doing.
underestimate decreasing negative (slope) to 0 to increasing positive (slope)
an object is speeding up if __________________ have the same sign an object is slowing down if __________________ have different signs
v(t) and a(t) v(t) and a(t)
the derivative of g is given by g'(x) = cos(4x^2) for (0,1.5). on what interval(s) is g decreasing
(0.627, 1.085) and (1.401, 1.5)
7. given the function h(x) = x^3 - 2x^2 + x, find the interval(s) when h is concave up and decreasing at the same time.
(2/3, 1)
11. a particle moves along the y-axis so that its velocity at time t, 0≤t≤6, is given by v(t) = 2(t - 2)(t - 5). find the minimum velocity of the particle.
-4.5
19. if the graph of y = x^3 + ax^2 + bx - 4 has a point of inflection at (1,-6), what is the value of b?
0
find lim x--> secx/cscx
0
1. what point on the graph y = √x is closest to (5,0) 2. two towers are 30 ft apart. one is 12ft, one is 28 ft high. stake on ground in between them with wire attack to each tower and stake. least amount of wire? 3. a particle is traveling along the x-axis and its position from the origin can be modeled by x(t) = t^3 - 15t^2 + 72t - 9 where x is cm and t is seconds. a. on the interval (3,9), find when the particle is farthest to the right. b. on the same interval, what is the particle's max speed?
1. (4.5, √4.5) 2. x = 3.264ft 3.a. 9 seconds b. 45cm/sec
1. you have 200 feet of fencing to enclose a rectangular field by your house. one side of the field borders a pond and does not require fencing. what is the area of the largest field you can enclose? 2. a rectangle has two vertices on the x-axis and two vertices on the curve of y = 9 - x^2. what is the maximum area of this rectangle? 3. a sheet of cardboard with side length 60 is folded into a rectangular box with a lid. four squares with side length x and two rectangular regions are discarded in order to fold the box, as shown. for what value of x is the volume of the box a maximum?
1. A = 5000ft^2 2. A = 20.785un^2 3. x = 10
JUST MAKING EQUATIONS (STEPS 1-3): 1. find two numbers whose sum is 30 and whose product is as large as possible 2. what point on the graph y = √x is closest to (5,0) 3. mr. Kelly needs to make an open-top box to store all his magic cards. he has a piece of cardboard that is 14 by 30 inches. to do this, mr. Kelly cuts out squares of equal size from the four corners so the four sides can be bent upwards. what size should the squares be in order to create a box with the largest possible volume? 4. two towers are 30 feet apart. one is 12 feet high and the other is 28 feet high. there is a stake in the ground between the towers. the top of each tower has a wire tied to it that connects to the stake on the ground. where should the stake be placed to use the least amount of wire?
1. P = 30x - x^2 2. d = √((x-5)^2 + x) 3. V = 420x - 88x^2 + 4x^3 4. W = √(28^2 + (30-x)^2) + √(144 + x^2)
write out the equation that needs to be "optimized." this equation should be in one variable. you do NOT need to solve the problem. we will solve in the next lesson. 1. what is the smallest product of two numbers given that one number is exactly 7 greater than the other number? 2. if the product of two positive numbers is 36, and the sum of the first number plus 4 times the second number is a minus, what are the two numbers? 3. a manufacturer wants to design an open box having a square base and a surface area of 108 square inches. what dimensions will produce a box with maximum volume? 4. Sullivan and brust are watching fireworks not he 4th of July. they build towers to get a better view, and decide to work together by securing them to the same stake in the ground. they place the towers 40 ft apart. mr. bursts tower is 10 feet high and mr. Sullivan's tower is 22 feet high. where should the stake be placed to use the least amount of wire? 5. which points on the graph of y = 4-x^4 are closest to the point (0,2)? 6. a rectangular page is to contain 24 square inches of print. the margins at the top and bottom of the page are to be 1.5 inches, and the margins on the left and right are to be 1 inch. what should the dimensions of the page be so that the least amount of paper is used? 7. you are creating an open-top box with a piece of cardboard that is 16 x 30 inches. what size of square should be cut out of each corner to create a box with the largest volume? 8. a rectangular pig pen using 300 ft of fencing is built next to an existing wall, so only three sides of fencing are needed. what dimensions should the farmer use to construct the pen with the largest possible area? 9. what is the radius of a cylindrical soda can with volume of 512 cubic inches that will use the minimum material? volume of a cylinder is V = πr^2h. surface area of a cylinder is SA=2πr^2 + 2πrh. 10. a power station is on one side of a river that is 1/2 mile wide, and a factory is 6 miles downstream on the other side. it costs $60,000 per mile to run power lines over land and $85,000 per mile to run them underwater. find the most economical path for the transmission line from the power station to the factory. hint: total cost = (on land cost)(distance on land) + (underwater cost)(distance in water) 11. a rancher has 200 feet of fencing with which to enclose two adjacent rectangular corrals. what dimensions should be used so that the enclosed area will be a maximum?
1. P = y^2 + 7y 2. M = x + 144/x 3. V = 27x - (x^3)/4 4. W = √(484 + (40-x)^2) + √(100 + x^2) 5. d = √(x^2 + (2-x^2)^2) 6. A = 3x + 48/x + 30 7. V = 480x - 92x^2 + 4x^3 8. A = 300y - 2y^2 9. A = 2πr^2 + 1024/r 10. C = 60000(6-x) + 85000(√(x^2 + 1/4)) 11. A = (400x - 8x^2)/3
1. use the function f(x) = -x^2 + 3x + 10 to answer the following. a. on the interval [2,6], what is the average rate of change? b. on the interval (2,6) when does the instantaneous rate of change equal the average rate of change?
1. a. -5 b. x = 4
1. a particle is traveling along the x-axis and it's position from the origin can be modeled by x(t) = -2/3t^3 + t^2 + 12t + 1 where x is meters and t is minutes on the interval. a. at what time t during the interval (0,4) is the particle farthest to the left? b. on the same interval what is the particle's max speed? 2. find the point on the graph of the function f(x) = x^2 that is closest to the point (2,1/2). 3. a particle moves along the x-axis so that any time t its position is s(t) = 1/3t^3 - 4t^2 + 7t - 5 where s is inches and t is hours. a. at what time t during the interval (0,6) is the particle farthest to the right? b. one the same interval what is the particle's max speed? 4. a rectangle is formed with the base on the x-axis and the top corners on the function y = 20-x^2. find the dimensions of the rectangle with the largest area. 5. what is the radius of a cylindrical soda can with volume of 512 cubic inches that will use the minimum material? 6. a swimmer is 500 meters from he closest point on a straight shoreline. she needs to reach her house located 2000 meters down shore from the closest point. if she swim 1/2m/s and she runs at 4m/s, how far from her house should she come ashore so as to arrive at her house in the shortest time? 7. mr. kelly is selling licorice for $1.50 per piece. the cost of producing each piece of licorice increases the more he produces. mr. Kelly finds that the cost to produce a piece of licorice is 10√x per piece where x is the number of licorice pieces. what is the most mr. Kelly could lose per piece not he sale of licorice. justify your answer.
1. a. t = 0 b. 12.5 meters/min 2. (1,1) 3. a. t = 1 b. 9 inches/hr 4. 40/3 X 2√(20/3) 5. r = 4.335 inches 6. 1937.006 meters 7. -$16.67
consider the curves in the xy-plane for each problem. at the given point, is the curve increasing or decreasing? justify your answer. 1. x^2 - y^2/2 = -1 at (-1,2) 2. x^(2/3) + y^(2/3) = 5 at (1,-8) 3. x^2 - 2xy + y^2 = 1 at (-1,-2)
1. decreasing because dy/dx<0 2. increasing because dyd/dx>0 increasing because dy/dx>0
strategies for solving optimization problems
1. draw picture (if applicable) and identify known and unknown quantities 2. write an equation (model) that will be optimized 3. write your equation in terms of a single variable 4. determine the desired max or min value with calculus techniques 5. determine the domain (endpoints) of your equation to verify if the endpoints represent a max or min
how do you find a critical point?
1. f'(x) does not exist f'(x) = 0
graph: x^2 + y^2 = 9 1. where does the reaction have horizontal and vertical tangent lines? 2. in which quadrant is the curve increasing? decreasing? 3. where is the curve concave up? concave down? 4. find dy/dx. 5. find d^2y/dx^2.
1. horizontal: x = 0 vertical: x = +/-3 2. increasing: Q2, Q4 decreasing: Q1, Q3 3. concave up: Q3, Q4 concave down: Q1, Q2 4. dy/dx = -x/y 5. -y^2 - x^2 / y^3
first derivative test is used for finding relative (local) extrema. to use:
1. identify critical points (f'(x) = 0 or DNE (undefined)) 2. make and label a "sign" chart by testing values between the critical values to determine if f'>0 or <0 3. state "f(x) has a relative max of _____ at x = ______ because f' changes from positive to negative" "f(x) have a relative min of _____ at x = ______ because f' changes from negative to positive"
find the relative extrema by using the second derivative test. justify your answer 1. f(x) = 5 + 3x^2 - x^3 2. h(x) = (2x - 5)^2 3. g(x) = x + 2sinx on the interval (0,2π) 4. f(x) = 2x^4 - 8x + 3
1. relative min at x = 0 because f'(0)=0 and f"(0)>0 relative max at x=2 b/c f'(2)=0 and f"(2)<0 2. absolute mina t x = 5/2 b/c h'(5/2)=0 and h"(5/2)>0 3. relative max at x = 2π/3 b/c g'(2π/3)=0 and g"(2π/3)<0 relative min at x = 4π/3 b/c g'(4π/3)=0 and g"(4π/3)>0 4. absolute min at x = 1 b/c f'(1)=0 and f"(1)>0
1. find two negative numbers that add up to -50 such that maximum product is possible 2. a t shirt maker estimate that the weekly cost of making x shirts is C(x) = 50 + 2x + x^2/200. the weekly revenue from selling x shirts is given by the function R(x) = 20x + x^2/200. a. derive the weekly profit function. b. what is the max weekly profit? 3. a company orders 600m of fence to enclose a rectangular area of their property against a straight river. the company only needs to fence in 3 sides. what is the maximum area that can be enclosed with these materials 4. find the point on the curve y = √x that is minimum distance from he point (4,0) 5. an open top box with a square bottom and rectangular sides is to have a volume of 256in^3. find the dimensions that require the minimum amount of material.
1. x = -25 y = -25 2. a. P(x) = 18x - 50 b. 274 3. A = 45,000m^2 4.(3.5,√3.5) 5. V = 8 x 8 x 4
let h be the function given by h(t) = 70 - 15cos(πt/3) + 5sin(πt/3) for (0,5). at what value of t is h increasing most rapidly?
1.343
10. let f(x) = xe^(-x) + ce^(-x), where c is a positive constant. for what positive value of c does f have an absolute maximum at x = -5? 11. let f(x) = 9-x^2 for x≥0 and f(x)≥0. an isosceles triangle whose base is the interval (0,0) to (b,0) has its vertex on the graph of f. for what value of b does the triangle have maximum area? recall that the area of a triangle is modeled by A = 1/2(base)(height) t(days): 0, 3, 8, 12, 20 A(t) (gallons): 2, 6, 9, 10, 7 a. use the data in the table to estimate the rate at which the number of gallons of apple juice in the tank is changing at time t = 10 days. show the computations that lead to your answer. indicate units of measure. b. for (0,12), is there a time t at which A'(t) = 2/3? justify your answer. c. the number of gallons of apple juice in the tank at time t is also modeled by the function B defined by B(t) = 3t - (1/2)(t + 4)^(3/2) + 6, where t is measured in days and (0,20). based on the model, at what time t, for (0,20), is the number of gallons of apple juice in the tank an absolute maximum? justify. d. for the function B defined in part c, the local linear approximation near t = 5 is used to approximate B(5). is this approximation an overestimate or an underestimate for the value B(5)?
10. c = 6 11. b = √12 12. a. 1/4 gallon/day b. yes b/c of the MVT on the interval [0,12] c. maximum on day 12 b/c B'(12) = 0 and B"(12)<0 d. overestimate b/c B"<0, which meant B(t) is concave down.
find the minimum sum of two positive numbers with a product of 50.
10√2
find the largest possible area of each object, given its boundaries. draw a picture to represent each problem. you don't need to solve. 12. a rectangle is formed in Q1 with one corner at the origin and the other corner on the line y = 8 - 2x. 13. a rectangle is formed with the base on the x-axis and the top corners not he function y = 6-x^2
12. A = 8x - 2x^2 13. A = -2x^3 + 12x
does the line tangent to the graph of h at the given value of x lie above or below the graph of h? why? 14. h(x) = 2x^3 - 4x^2 - 3x at x = -2 15. h'(x) = (x^2 - 4)/x at x = 2
14. above because h"(-2)<0 15. below because h"(2)>0
while camping on Conway lake, Matt spots a fire burning in a neighboring lean-to. Matt grabs his bucket and heads to the lake so he can fill his bucket and douse the fire. his tent is 36 meters from the lake and the lean-to is 64 meters from the water. he can run 8 meters per second with an empty bucket and 4 meters per second with a full bucket how long does it take Matt to get to the fire?
25.369 seconds
for each function, find the intervals where it is increasing and decreasing, and justify your conclusion. construct a sign chart to help you organize the information, but do not use a calculator. 3. f(x) = x^3 - 12x + 1 4. g(x) = x^2(x - 3) 5. f(x) = x^2e^x 6. g(t) = 12(1 + cost) on the interval (0,2π)
3. increasing on (-∞, -2) and (2,∞) because f'(x)>0 decreasing on (-2,2) because f'(x)<0 4. increasing on (-∞, 0) and (2,∞) because f'(x)>0 decreasing on (0,2) because f'(x)<0 5. increasing on (-∞,-2) and (0,∞) because f'(x)>0 decreasing on (-2,0) because f'(x)<0 6. increasing on (π, 2π) because f'(x)>0 decreasing on (0,π) because f'(x)<0
the first derivative of the function f is given by f'(x) = -2 + x + 3e^(-cos(4x)). how many points of inflection does the graph of f have on the interval [0,π]?
4 points
consider the given differential equation dy/dx, where y = f(x) is a particular solution with a given point. for each problem, determine if f has a relative minimum, a relative maximum, or neither at the given point. justify. 4. dy/dx = ysinx where f(2π) = 1 5. dy/dx = x/y + lnx where f(1) = -2 6. dy/dx = yx^2 where f(0) = -5
4. relative minimum b/c dy/dx = 0 and d^2y/dx^2>0 5. neither dy/dx does not equal 0 6. neither. d^2y/dx^2 = 0 therefore f is not concave up nor down.
find the critical points 4. f(x) = 4x^3 - 9x^2 - 12x + 3 5. g(t) = 2/(t^2 - 4) 6. h(x) = ^3√(x-2) 7. f(x) = (lnx)^2 8. h(x) = 2sin(x/2) where -2π<=<=2π 9. g(x) = e^x - x
4. x = -1/2, x = 2 5. t = 0, t = +/-2 6. x = 2 7. x = 1 8. x = -π, x = 0, x = π 9. x = 0
state the intervals of concavity and justify your answer. 6. g(x) = x/(x-1) 7. f(x) = x^3 - 12x
6. concave down on (-∞,1) because g"<0 concave up on (1,∞) because g">0 7. concave down on (-∞, 0) because f"<0 concave up on (0,∞) because f">0
13. find the maximum acceleration attained on the interval 0≤t≤3 by the particle whose velocity is given by v(t) = 2/3t^3 - 4t^2 + 8t - 2
8
look at page #2 of 5.4 for the two problems with graphs
YUH!!!!!
go to the top of the first page of 5.3 to do the graph problem
YUH!!!!!!
look at page 2 of 5.6 for the graph problems
YUH!!!!!!
look at the problem in the middle of the page on page #2 of 5.3 for the graph
YUH!!!!!!
look at first page of 5.5 for graph!! IMPORTANT!!!!
YUH!!!!!! this considers how much is going up vs how much is going down
look at page 3 of 5.5 to find graphs on the top
YUH!!!!!!!
look at page 4 of 5.6 for the six graphs
YUH!!!!!!!!
look at unit 5 extra practice on pages 2 and 3 for graphs
YUH!!!!!!!!!
look at unit 5 review-analytical applications of differentiation for a beginning overview
YUH!!!!!!!!!!!!!$$$$$$$$$$$!!!!!!!!!!!!$$$$$$$$$$$$$
14. let f be the function given by f(x) = 4 - x. g is a function with derivative given by g'(x) = f(x)f'(x)(x - 2). on what intervals is g decreasing?
[2,4] only
a particle is traveling on the y-axis and its position from he origin can be modeled by y(t) = 6t - 2t^3 + 10, where y is meters and t is minutes a. on the interval (0,2), when is the particle farthest above the origin. b. on the interval (0,2) what is the particle's max speed?
a. at t = 1 minutes b. 18 meters/min
3. f(x) = -xe^(x/4)
absolute max at x = -4 b/c f'(-4)=0 and f"(-4)<0
what is the absolute maximum value of the function g(x) = 2x^3 + 3/2x^2 - 3x - 10 on the closed interval [-2,2].
absolute maximum value of 6
9. let f be the function defined by f(x) = cos^2x - cosx for 0≤x≤3π/2. find the absolute maximum value and the absolute minimum value of f.
absolute minimum value is -1/4 absolute maximum value is 2
global vs local extrema aka
absolute vs relative extrema look at page #1 of 5.2 to find first graph and problems 1 and 2
find d^2y/dx^2 in terms of x and y for 1 - xy = x - y
d^2y/dx^2 = (2y + 2)/(x^2 - 2x + 1)
the second derivative test: suppose f'(c) = 0, then: if f"(c)>0, then... if f"(c)<0, then...
f has a relative minimum at x = c f has a relative maximum at x = c
find the interval(s), if any, on which the function y = 1/x^2 is increasing and decreasing. justify your answer
increasing on the interval (-∞,0) because y'>0 decreasing on the interval (0,∞) because y'<0
points of inflection on a graph of f are
like in between min and max and are equal to 0 when f"(x)
2. find the relative max/min of the function h(x) = x^2/(4-x)
min at x = 0 b/c f' changes sign from neg to pos max at x = 8 b/c f' changes sign from pos to neg
use the first derivative test to help you find the relative minimum value of f(x) = xlnx. what is this value?
min value is -1/e
when the graph is of f'(x), points of inflection are... when graph is of f"(x), points of inflection are..
mins, maxes, cusps, or corners zeros
if there is only ______ critical point, and that CP is an extremum (max or min), then it is an ___________ extremum (max or min)
one absolute
the critical points give us... the ends of the interval (the endpoints), give us....
possibilities of finding a max or min. other possibilities of dinging a max or min.
locate the x values of all extrema of f(x) = 2 + 2x^2-x^4. classify each value as a relative max or min. justify.
relative max at x = -1 and x = 1 because f' changes sign from positive to negative relative min at x = 0 because f' changes sign from negative to positive
1. use the second derivative test to find the relative extrema of f(x) = x^4 - 2x^2
relative max at x = 0 b/c f'(0) = 0 and f"(0)<0. relative min at x = -1 b/c f'(-1) = 0 and f"(-1)>0 relative min at x = 1 b/c f'(1) = 0 and f"(1)>0
use the 2nd derivative test to find the x-value(s) of all relative extrema of the function f(x) = 2sinx + √(2)x on the interval [0,2π]. justify your answer.
relative max at x = 3π/4 because f'(3π/4) = 0 and f"(3π/4)<0 relative min at x = 5π/4 because f'(5π/4) = 0 and f"(5π/4)>0
2. use the second derivative test to find the relative extrema of f(x) = √(2x) - 2cosx on the interval [0,2π]
relative min at x = 7π/4 b/c f'(7π/4) = 0 and f"(7π/4)>0 relative max at x = 5π/4 b/c f'(5π/4) = 0 and f"(5π/4) <0
if you want to know if something is increasing or decreasing, your look at the...
sing of its rate of change
focus on the ______ of f. the ________ of f is the _________ of f'
slope slope y-value
1. a particle's position along the x-axis is measured by x(t) = 1/3t^3 - 3t^2 + 8t + 1 where t>0. find the intervals where the particle is speeding up. find the intervals where the particle is slowing down.
speeding up on (2,3) and (4,∞) b/c a(t) and v(t) have the same sign. slowing down on (-∞,2) and (3,4) b/c v(t) and a(t) have different signs
2. a particle is moving along the x-axis with position function x(t) = 1/3t^3 - 4t^2 + 12t. find the velocity and acceleration functions. describe the motion of the particle.
speeding up on (2,4) and (6,∞) b/c v(t) and a(t) have the same sign slowing down on (-∞,2) and (4,6) b/c v(t) and a(t) have different signs
12. a particle moves along the x-axis with position at time t given by x(t) = e^(-t)cost for 0≤t≤2π. find the time t at which the particle is farthest to the right.
t = 0
8. a particle moves along the x-axis so that its position at any time t>=0 is given by x(t) = t^3 - 3t^2 + t + 1. for what values of t, 0<=t<=2, is the particle's instantaneous velocity the same as its average velocity on the closed interval [0,2]?
t = 0.423 t = 1.577
find all the critical points of y = 2x^3 + 3x^2 - 36x. justify your answer
the critical points are at x = 2 and x = -3 because they are zeros of the derivative
consider the curve defined by x^2/16 - y^2/9 = 1. it is known that dy/dx = 9x/16y and d^2y/dx^2 = -81/16y^3. which of the following statements is true about the curve in Q4
the curve is concave up because d^2y/dx^2 > 0
let f be a polynomial function with values of f'(x) at selected values of x given in the table above. which of the following must be true for [-5,3] x: -5, -4, -2, 0, 3 f'(x): -8, -10, -7, -4, -6
the graph of f has at least two points of inflection
which of the following statements about the function given by f(x) - x^4 - 2x^3 is true?
the graph of the function has two points of inflection and the function has one relative extremum
on a graph of f', f is increasing when f is decreasing when
the line is above the x-axis below the x-axis
the sign of a rate of change can tell you if the dependent variable is increasing or decreasing. interpret the following: students/yr>0 miles/hr<0 mastery checks/week>0 virus cases/month^2<0
the number of students is increasing; the number of miles is decreasing; the number of mastery checks is increasing; the rate of virus cases per month is decreasing
on a f' graph, the maxes and mins are at....
the x-axis (like when the graph cross the x-axis)
justification statements: assume c and d are critical numbers of a function f
there is a minimum value at x = c because f' changes signs from negative to positive there is a maximum value at x = d because f' changes signs from positive to negative
point of inflection
there is a point of inflection of f at x = c if f(c) is defined and f" changes sign at x = c. in other words, a point of inflection is where the graph changes concavity.
the first derivative of the function f is given by f'(x) = cos^2x/x - 1/5. how many critical values does f have on the open interval (0,10)?
three
optimize
to make the best or most effective use of a situation or resource
10. the first derivative of the function f is given by f'(x) = (sin^2x)/x - (2/9). how many critical values does f have on the open Interval (0,10)?
two
points of inflection on f(x) = sinx
when the line cross the x-axis
12. let f be the function given by f(x) = 2sinx. as shown above, the graph f crosses the origin at point A and point B at the coordinate point (π/2, 2). find the x-coordinate of the point on the graph of f, between points A and B, at which the line tangent to the graph of f is parallel to line AB. round or truncate to three decimals.
x = 0.881
find the point at which the graph of 4x^2 + y^2 - 8x + 4y + 4 = 0 has a vertical or horizontal tangent line.
x = 1 y = -2
the function f has first derivative given by f'(x) = √x - e^x/x. what is the x-coordinate of the inflection point of the graph of f?
x = 1.198 bc f"(x) = 0 at this point and f"(x) changes sign from pos to neg
apply the mean value theorem to y = x^3 - 2x^2 - 3 to find when the instantaneous rate of change will equal the average rate of change on the interval [0,2]
x = 4/3
in graph f'(x): f'>0 and f is increasing if it is above the
x-axis
first derivative test chart set up
x: intervals and critical points f(x): positive or negative, and 0 or undefined
look at all of 5.8 for sketching graphs of derivatives
yuh!!!!!!!!!!!!!!!!!!!!!!!!!!!$$$$$$$$$$$$$$$$$$$$$$$
1. x: (-3, -1/2), -1/2, (-1/2, 3) g''(x): positive, 0, negative use the table above to find the following. intervals where g(x) is concave up: intervals where g(x) is concave down: points of inflection:
(-3, -1/2) (-1/2, 3) x = -1/2
at a top or bottom of the function, where it changes from positive to negative or negative to positive slope, f' = ? when it is rounded when it is pointed f' =
0 undefined
leanne is driving east on Reagan memorial tollway in Illinois. two weeks later she receives a speeding ticket in the mail and a print-out of her E-ZPass records. toll plaza: rock falls, maple park, elmhurst mile marker: 45, 100, 140 time stamp: 1:23 pm, 2:11 pm, 2:40 pm 1. how many hours did it take leanne to make it from rock falls to maple park? 2. what was her average speed between rock falls and maple park? 3. if the speed limit is 70mph, is it possible that leanne was speeding during that time period? explain. 4. do police officers have convincing evidence from the E-ZPass alone that Leanne was speeding somewhere between Rock falls and maple park? 5. how many hours did it take leanne together from the maple park plaza to the Elmhurst plaza? 6. what was her average speed between maple park and elmhurst? 7. do police officers have conniving evidence from the E-ZPass alone that leanne was speeding somewhat between maple park and elmhurst? how do you know? 8. must there have been a moment when leanne was driving exactly at her average speed? how do you know?
1. 0.8 hours 2. 68.75mph 3. yes it is possible that she was speeding because 70 mph is only 1.25 mph from 68.75 mph, and there has to be values above and below the mean at which she was going in order to reach this mean, therefore she could have been goin 70 mph, as it is close. 4. no recuasse it is possible that she drove at a constant rate the entire time, and the E-ZPass will not give this information. 5. 0.967 hours 6. 82.76 mph 7. yes, if on average she drove 82.76 mph, then she must have driven about 70 mph at one point. 8. yes, even if she drove mostly above or below 82.76mph, she couldn't just skip over 92.76mph distance is a continuous and differentiable function. at some movement, her instantaneous rate of change must have been the same as her average ROC.
find the absolute maximum value and the absolute minimum value of the function on the given interval. 1. f(x) = 1 + (x + 1)^2, [-2,5] 2. f(x) = 2x^3 + 3x^2 + 4, [-2,1] 3. f(x) = x / (x^2 + 1), [-2,2] 4. f(x) = sin(x + π/4), [0,7π/4] 5. g(x) = xe^(2x), [-1,1] 6. f(x) = x^3 + 2x^2 + x - 5, [-2,2]
1. absolute max value is 37; absolute min value is 1 2. absolute min value is 0; absolute max value is 9 3. absolute min value is -1/2; absolute max value is 1/2 4. absolute min value is -1; absolute max is 1 5. absolute min value is -0.184; absolute max value is 7.389 6. absolute min is -7; absolute max is 13
two common mistakes when finding a point of inflection
1. assuming that f" = 0 means there is a point of inflection 2. assuming that f" does not equal 0 (DNE) means there is no point of inflection point of inflection at a cusp/corner: yes a point of inflection but f' and f" don't exist
how to prove with MVT:
1. check conditions 2. find aver ROC on [a,b] 3. make conclusion using MVT, incorporating question stem
find the absolute maximum value and the absolute minimum value of the function on the given interval. remember to show that you checked all the candidates 1. f(x) = x^3 - 27x + 2, [0,4] 2. h(x) = 3x^(2/3) - 2x, [-1,1] 3. g(x) = x^2 + (2/x), [1/2,2] 4. f(x) = sin3x, [-π/2,π/2]
1. min value: -52, max value: 2 2. min value: 0, max value: 5 3. min value: 3, max value: 5 4. min value: -1, max value: 1
skater sully is riding a skateboard back and forth on a street than runs north/south. the twice-differentiable function S models Sully's position on the street, measured by how many meters north he is from his starting point, at time t, measured in seconds from the start of his ride. the table below gives the values of S(t) and Sully's velocity v(t0 at selected times. t seconds: 0, 4, 10, 20 S(t) meters: 0, 20, 32, -10 v(t) meters per second: 0, 2.5, -2.3, -0.5 1. for 4<=t<=10, must there be a time t when Sully is 30 meters from his starting point? justify your answer. 2. for 0<=t<=4, must there be a time t when Sully's velocity is 2.7 meters per second? justify your answer.
1. yes because of the IVT 2. No, neither IVT or MVT will prove this.
10. the rate of money brought in by a particular mutual fund is represented by m(t) = (e/2)^t thousand dollars per year where t is measured in years. is the amount of money from this mutual fund increasing or decreasing at time t = 5 years? justify your answer. 11. the number of hair follicles on mr. Sullivan;s scalp is measured by the function h(t) = 500e^(-t) where t is measured in years. is the amount of hair increasing or decreasing at t = 7 years? justify your answer. 12. the rate at which rainwater flow into a street gutter is modeled by the function G(t) = 10sin(t^2/30) cubic feet per hour where t is measured in hours and 0≤t≤8. the gutter's drainage system allows water to flow out of the gutter at a rate modeled by D(t) = -0.02x^3 + 0.05x^2 + 0.87x for 0≤t≤8. is the amount of water in the gutter increasing or decreasing at time t = 4 hours? give a reason for your answer.
10. increasing because m(5)>0 11. decreasing because h'(7)<0 12. increasing because G(4)-D(4)>0
find the point(s) of inflection for each function. justify your answer. 2. f(x) = sin(x/2) on the interval (-π, 3π) 3. f(x) = e^(-x)^2 4. h(x) = (2x^2 - 5)^2 5. f(x) = 2x^4 - 8x + 3
2. points of inflection at x = 0 and x = 2π because f'' changes sign. 3. points of inflection at x = +/-√(1/2) because f" changes sign 4. points of inflection at x = -√5 and √5 because h" changes sign 5. no points of inflection because f"(x) does not change sign
find all critical points: 3. f(x) = 1/3x^3 - 9x + 24 4. g(x) = 1/(√(4-x^2))
3. x = +/-3 4. x = 0, x = +/-2 *these are only candidates for being extrema, it does not mean that they are extrema*
use the mean value theorem, find where the instantaneous rate of change is equivalent to the average rate of change. 3. y = x^2 + 6x + 9 on [-4,-2] 4. y = -√(7x + 21) on [-3,1] 5. y = (-x^2 + 1)/(4x) on [1,3]
3. x = -3 4. x = -2 5. x = √3
find the critical points of each function. 3. g(x) = x^2e^x 4. f(x) = cos(πx) where -π<=x<=π
3. x = 0, x = -2 4. x = 0, +/-1, +/-2, +/-3
for each problem, the derivative of a function g is given. find all relative max/min of g and justify. 4. g'(x) = (x + 4)e^x 5. g'(x) = x^2 + 5x + 4
4. minimum at x = -4 because g' changes sign from negative to positive 5. maximum at x = -4 because g' changes sign from pos to neg minimum at x = -1 because g' changes sign from neg to pos
using the MVT, find where the instantaneous rate of change is equivalent to the average rate of change 4. y = x^2 - 5x + 2 on [-4,-2] 5. y = sin3x on [0,π] 6. y = (-5x + 15)^(1/2) on [1,3] 7. y = e^x on [0, ln2]
4. x = -3 5. x = π/6 x = π/2 x = 5π/6 6. x = 2.5 7. x = 0.367
use a calculator to help find all x-values of relative max/min of f. no justification necessary. 6. f'(x) = x^3 - 6cos(x^2) + 2 7. f'(x) = (2 - lnx) / x^2 8. f'(x) = √(x^4 + 2) + x^2 - 5x
6. min at x=-1.922 and x = 1.018 max at x = -1.250 7. max at x = 7.389 8. min at x = 2.467 max at x = 0.302
use the first derivative test to locate the x-values of all extremes. classify if it is a relative max or min and justify your answer. 9. f(x) = x^3 - 12x + 1 10. g(x) = xe^(5x) 11. h(x) = x^3 / (x + 1) 12. ??? 13. what is the maximum value of g(x) = 2cosx on the open interval (-π,π)? 14. what is the relative minimum value of h(x) = -x^3 + 6x^2 - 3?
9. max at x = -2 b/c f'(x) changes sign from pos to neg min at x = 2 b/c f'(x) changes sign from neg to pos 10. min at x = -1/5 because g'(x) changes sign from neg to pos 11. min at x = -3/2 because h'(x) changes sign from neg to pos 13. 2 14. -3
13. a differentiable function g has the property that g'(x)>2 for 1<=x<=5 and g(4) = 3. which of the following could be true? I. g(1) = -6 II. g(2) = 0 III. g(5) = 4
I only
we use the _____ to justify conclusions about a function over an interval
MVT
4. the rate of change of fruit flies in mr. Kelly's kitchen at time t days is modeled by R(t) =2tcos(t^2) flies per day. show that the number of flies is decreasing at time t = 3.
R(3) = -5.467 decreasing because R(3)<0
SHOW ALL CANDIDATES
YUH!!!!!!
look at page #2 for 5.2 to find 1, 2, and 3 of the practice not the bottom of the page
YUH!!!!!!
look at the top of page #3 of 5.3 for the two graph problems
YUH!!!!!!!!!!
15. particle X moves along the positive x-axis so that its position at time t≥0 is given by x(t) = 2t^3 - 7t^2 + 4. a. is particle x moving towards the left or towards the right at time t = 2? give a reason for your answer. b. at what time t≥0 is particle X farthest to the left? justify your answer. c. a second particle, y, moves along the positive y-axis so that its position at time t is given by y(t) = 4t + 5. at any time t, t≥0, the origin and the positions of the particles x and y are the vertices of a rectangle in the first quadrant. find the rate of change of the area of the rectangle at time t = 2. show the work that leads to your answer.
a. moving left because x'(t) < 0 b. at t = 7/3 the particle moves right for t>7/3 c. A'(2) = -84
3. a hot air balloon is launched into the air with a human pilot. the twice-differentiable function h models the balloon's height, measured in feet, at time t, measured in minutes. the table below gives values of the h(t) and the vertical velocity v(t) of the balloon at selected times t. t minutes: 0, 6, 10, 40 h(t) feet: 0, 46, 35, 105 v(t) feet per minute: 0, 6, 20, 5 a. for 6<=t<=10, must there be a time t when the balloon is 50 ft in the air? justify your answer. b. for 10<=t<=40, must there be a time t when the balloon's velocity is 3 feet per second? justify your answer.
a. no, the IVT shows it must be between 46 ft and 35 ft b. yes, because the IVT for the values of velocity. the MVT does not prove a 3ft/sec velocity.
1. find the intervals on which the function f(x) = -x^2 - 4x - 1 is increasing and decreasing and justify your answers. a. first find the critical points b. in between the x-values, the derivative must be positive or negative c. we can use a chart to help keep track of the information. write the critical points of the derivative first. d. answer statements with justification:
a. x = -2 b. c. x: (-infinity, -2), -2, (-2, infinity) sign of f'(x): positive, 0, negative d. f is increasing on (-infinty, -2) because f'(x)>0 f is decreasing on (-2, infinity) because f'(x)<0!
2. t minutes: 0, 5, 15, 20, 30 h(t) feet: 0, 40, 70, 65, 80 v(t) feet per minute: 0, 10, 3, 2, 4 a hot air balloon is launched into the air with a human pilot. the twice-differentiable function h models the balloons height, measured in feet, at time t, measured in minutes. the table above gives values of the h(t) and the vertical velocity v(t) of the balloon at selected times t. a. for 5 <= t <= 20, must there be a time t when the balloon is 50 feet in the air? justify your answer. b. for 20<=t<=30, must there be a time t when the balloon's velocity is 1.5 feet per second? justify your answer.
a. yes because of the IVT b. yes because of the MVT
1. skater sully is riding a skateboard back and forth on a street that runs north/south. the twice-differentiable function s models sully's position on the street, measured by how many meters north he is from his starting point, at time t, measured in seconds from the start of his ride. the table below gives values of the S(t) and Sully's velocity v(t) at selected times t. t seconds: 0, 20, 30, 60 S(t) meters: 0, -5, 7, 40 v(t) meters per second: 0, 3.2, 0.8, -0.9 a. for 0<=t<=20, must there be a time t when Sully is 2 meters south of his starting point? justify your answer. b. for 30<=t<=60, must there be a time t when Sully's velocity is 1.1 meters per second? justify your answer.
a. yes because of the IVT b. yes, because of the MVT
2. a particle is moving along the x-axis. the twice-differentiable function s models the particles distance from the origin, measured in centimeters, at time t, measured in seconds. the table below gives values of the s(t) and the velocity v(t) of the particle at selected times t. t seconds: 3, 10, 20, 25 s(t) cm: 5, -2, -10, 8 v(t) cm per second: -4, -2, 3, -2 a. for 20<=t<=25, must there be a time t when the particle is at the origin? justify answer. b. for 3<=t<=10, must there be a time t when he particle's velocity is -1.5cm per second? justify your answer.
a. yes, because of the IVT b. yes, because of the MVT
4. does the line tangent to the graph of f(x) = ex^(-x) at x = 1 lie above or below the graph of f? why?
above because f''(1) > 0
if h(c) does not exist, then x = c...
cannot be a critical point
f''>0 means f is... f''<0 means f is...
concave up concave down look at graph on page 1 of 5.6 for more description
1. find the intervals of concavity for f(x) = 1/4x^4 - 6x^2 + x - 3
concave up on (-∞, -2) and (2,∞) because f">0. concave down on (-2,2) because f''<0. points of inflection at x = -2 and x = 2 because f'' changes sign
15. if g is a differentiable function such that g(x) < 0 for all real numbers x and if f'9x) = x^2 - x - 12)g(x), which of the following is true?
f has a relative minimum at x = -3 and a relative maximum at x = 4
2. find the intervals on which the function f(x) = 1/3x^3 - x^2 - 15x + 2 is increasing and decreasing and justify your answers.
f is increasing on (-infinity, -3) and (5, infinity) because f'(x)>0. f is decreasing on (-3,5) because f'(x)<0.
13. x: 1, 2, 3, 4, 5 f(x): -6, -1, 3, 6, 8 the table above gives values of a function at selected values of x. if f is twice-differentiable on the interval 1≤x≤5, which of the following statements could be true?
f' is positive and decreasing for 1≤x≤5
start with something we know. a quadratic function's graph is a parabola (the graph that is shown has a concave up parabola, no stretch/shrink, and the center/middle point is (2,-6)). we know f(x) = x^2 - 4x - 2 opens up, so f will have a minimum. examine the graph of this parabola nd describe the behavior of f'(x) around the minimum
f' neg to 0 to pos concave up at 0, cusp/corner when changed to positive f' pos to CP to neg max at critical point, cusp/corner when changed to negative
11. if f is a continuous, decreasing function on [0,10] with a critical point at (4,2), which of the following statements must be false?
f'(4)<0 the derivative must be zero or does not exist. it can't be negative if the point is a critical point.
if a function f(x) is continuous on [a,b] and differentiable on (Abraham), then there exists a value c for a<c<b such that...
f'(c) = f(b)-f(a) / b-a
"what is the maximum value" is not the same as "where is the maximum value"
f(c) x = c
find the relative maximums and minimums of f(x) = x^3 + 3x^2 - 9x + 5
f(x) has a relative max at x = -3 because f' changes from positive to negative f(x) has a relative min at x = 1 because the derivative changes from negative to positive
extreme value theorem
if a function f is continuous over the interval [a,b], then f has at least one minimum value and at least one maximum value of [a,b].
1. assume f(x) is continuous for all real numbers. the sign of its derivative is given in the table below for the domain of f. identify all relative extrema and justify your answers. interval: (-∞, -2), (-2,0), (0,3), (3,∞) f(x): positive, negative, negative, positive
max at x = -2 b/c f' changes from pos to neg min at x = 3 b/c f' changes from neg to pos
MVT vs IVT definitions:
mean value theorem: the derivative (instantaneous rate of change) must equal the average rate of change somewhere in the interval. intermediate value theorem: on the given interval, you will have a y-value at each of the end points of the interval. every y-value at least once in the interval.
1. use the First Derivative Test to find the x-values of any relative extrema of f(x) = (x^2 - 4)^(2/3)
minimum at x = -2 and x = 2 because f' changes sign from neg to pos maximum at x = 0 because f' changes sign from pos to neg
14. let f be the function with f(1) = e, f(4) = 1/e, and the derivative given by f'(x) = (x-1)(sin(ex)). how many values of x in the open interval (1,4) satisfy the conclusion of the MVT for the function f on the closed interval [1,4]?
more than two
min = max =
neg to pos pos to neg
2. a test plane flies in a straight line with positive velocity v(t0, in miles per minute, at time t minutes, where v is a differentiable function of t. selected values of v(t) are shown in the table below. t minutes: 0, 5, 10, 15, 20, 25, 30, 35, 40 v(t) miles per minute: 7.0, 9.2, 9.5, 7.0, 4.5, 2.4, 2.4, 4.3, 7.3 based on the values in the table, what is the smallest number of instances at which the acceleration of the plane could equal zero on the open interval 0<c<40? justify your answer.
since v(t) is differentiable and continuous, the MVT guarantees at least 2 instances where v'(t) = a(t) = 0 since there are 2 intervals (0,15] and [25,30] for which the average ROC is 0.
10. let g be a continuous function. the graph of the piecewise-linear function g', the derivative of g, is shown above for -4<=x<=4. find the average rate of change of g'(x) on the interval -4<=x<=4. does the mean value theorem applied on the interval -4<=x<=4 guarantee a value of c, for -4<x<4, such that g''(c) is equal to this average rate of change? why or why not?
undefined no, because g'(x) is not differentiable. it has several corners. the MVT only applied if the function is differentiable.
