AP Calc AB (Serrano) 2.6 Notes
How is the second derivative of y written in Leibowitz notation?
(d/dx)(dy/dx) = d²y/dx²
how do you denote the second derivative of f (the derivative of a derivative)?
(f')' = f² **superscript NOT exponent
Solve the following: (√x - 3)/(x-9)
(√x - 3)/(x-9) * (√x + 3)/(√x + 3) (x-9)/(x-9) cancels so = 1/(√x + 3)
Find the following limit by showing all algebraic work: lim(x→-∞) (2x + 5)/√(x²+4)
* (-1/x)/(1/√x²) (negative bc negative infinity) = (-2 - 5/x)/√(1 + 4/x²) -2/√(1+0) = -2
Describe (d/dx)[f(x)]: - name - function
- "d/dx of f at x" or "the derivative of f at x" - emphasizes the idea that differentiation is an operation performed on x
Describe df/dx: - name - function
- "the derivative of f with respect to x" - emphasizes the name of the function
Describe dy/dx: - name - function
- "the derivative of y with respect to x" - names both variables and uses d for derivative
Describe y': - name - function
- "y prime" - brief but does not name the independent variable
What are the values for the following problems (need to memorize): - lim(x→0) sin x/x - lim(x→∞) sin x/x
- = 1 - = 0
What does the second derivative tell us about the original function?
- concavity - points of inflection
How do you find the limit as x→±∞?
- determining VA and HA from given limits - graphing/sketching based off of given limits
What graphs should you know how to sketch?
- f' from graph of f - f from graph of f' - graph of f given conditions of f' and f"
What five ways can denote the derivative of a function y = f(x)?
- f'(x) - y' - dy/dx - df/dx - (d/dx)[f(x)]
What do the following on f(x) mean for f'(x) and f"(x): - concave up - concave down
- f'(x) increasing, f"(x) positive - f'(x) decreasing, f"(x) negative
What does it mean if f''(x) < 0 on an interval?
- f'(x) is decreasing - f(x) is concave down in this interval
What does it mean if f''(x) > 0 on an interval?
- f'(x) is increasing - f(x) is concave up in this interval
Describe f(x) for the following given f'(x) on an interval: - f'(x) > 0 - f'(x) < 0 - f'(x) = 0 (or a)
- f(x) is increasing on the interval - f(x) is decreasing on the interval - f(x) has a horizontal tangent at the x-value
Given the graph y = f'(x) where it is negative from (-1,1), positive from (-∞,-1)u(1,∞), appears to have an asymptote at y = 1, has two zeroes at x = -1,1 and has a turning point at x = 0, what can you say about f(x) given the point f(0) = 0?
- graph is increasing from (-∞,-1)u(1,∞) and decreasing from (-1,1) - turning points at x = -1 and 1 - steepest at x = 0 (point of inflection) - x = -1 is local max and x = 1 is local min - concave up from (-∞,0) and concave down from (0,∞)
What does the first derivative tell us about the original function?
- increasing/decreasing - max/min/rest
What are the three types of discontinuity?
- jump - removable - infinite (asymptote)
What are the three types of horizontal tangents you can have?
- maximum - minimum - uphill/downhill rests (2 uphill or downhill slopes that are briefly horizontal before continuing on)
Consider the following function and its derivatives. How do the graphs differ? - f(x) = x³ - x - f'(x) = 3x² - 1 - f''(x) = 6x
- negative slope between two turning points for the original function has its steepest point halfway between them, at x = 0; location of turning point for first derivative and transition from - to + for second derivative - switches from below to above x axis at zeroes
Describe what the following for slope on f(x) mean for f'(x): - increasing - decreasing
- positive - negative
What values are possibly present if f"(x) = 0?
- possible local max of f'(x) - possible local min of f'(x) - possible uphill rest of f'(x) - possible downhill rest of f'(x)
What two things should the derivative be interpreted as?
- the slope of the tangent line - a rate of change
How should you evaluate limits as x→c if direct substitution does not work?
a) factoring b) rationalize by multiplying by the conjugate c) limit laws d) squeeze theorem e) clear complex fractions
When is the function f differentiable on an open interval (a,b) (or (a,∞), (-a,∞), or (-∞,∞))?
if it is differentiable at every number in the interval
describe velocity
rate of change; find slope
What is the acceleration function?
s''(t) = v'(t) = a(t)
What is the velocity function?
s'(t) = v(t)
What is the position function?
s(t)
What techniques can you use to find F from f?
same as finding f(x) from f'(x)
How can we write the equation of the tangent line to the curve y = f(x) at the point (a, f(a)) in point slope form?
y - f(a) = f'(a)(x-a), where the derivative is the slope value at x = a
What do you know once you have a derivative function f'(a)? What can you do once you know it?
you know the slope; you can write hte equation of the tangent line at (a, f(a))
What does a local minimum or maximum for f(x) look like for f'(x)?
zeroes where f'(x) changes signs
What should you do if something is not differentiable at a point?
give a reason why
For the function f(x) = x³ - x, compare the graphs of f and f'
graph of f(x) = x³ - x: turning points at x = ± 0.6, passes through the origin, negative slope between turning points and positive slope outside of them (-∞,-0.6)u(0.6,∞) graph of f'(x) = 3x²-1: parabola with a 0 or turning point at (0,0.6); negative between (-0.6,0.6) on the x axis and positive outside of it (-∞,-0.6)u(0.6,∞)
What is the distance between a and the x coordinate?
h
Describe a possible graph of the function f(x) that fulfills the following conditions: i) f'(x) > 0 on (-∞,1), f'(x) < 0 on (1,∞) ii) f"(x) > 0 on (-∞,-2) and (2,∞); f"(x) < 0 on (-2,2) iii) lim(x→-∞) f(x) = -2, lim(x→∞) f(x) = 0
i) f'(x) > 0 (-∞,1) f'(x) < 0 (1,∞) + to - so local max ii) f"(x) > 0 on (-∞,-2) and (2,∞) = concave up; f"(x) < 0 on (2,-2) = concave down - turning points -2,2 iii) horizontal asymptotes at -2 and 0
What is a very important theorem for derivatives?
if f is differentiable at a, then f is continuous at a
When is a function f differentiable at a?
if f'(a) exists
What happens if you can find the derivative at a point?
the graph must continuous at the point
What are: - anything approaching the point a with a slope of ∞ or -∞ (ex. lim(x→3⁺) f(x) = -∞) - anything approaching ∞ or -∞ with a slope of a (ex. lim(x→∞) f(x) = -1
- vertical asymptote - horizontal asymptote
Describe your answers for the following if you are given a velocity v. time graph: - when was the car stopped? - when was the car accelerating and decelerating? - what happened at __ points in terms of acceleration?
- when the graph touches the x axis - when there is a slope (positive = accelerating, negative = decelerating) - straight line = no acceleration
What are the three ways a function can fail to be differentiable at a point a and why? Describe graphs as needed
1) cusp/corner (can't find derivative at a) ex. a curved graph that is pinched in the middle 2) a jump discontinuity (can't draw tan line) 3) straight line up and down (undefined); vertical tangent lines don't work ex. a graph with two curved slopes connected by a vertical drop at point a
A manufacturer produces bolts of fabric with a fixed width. The cost of producing x yards of this fabric is C = f(x) dollars. 1) What is the meaning of f'(x)? What are its units? 2) What does it mean to say that f'(1000) = 9?
1) f'(X) means the rate of change of the cost to produce fabric with respect to the number of yards produced. units = $/yd 2) When 1000 yards are produced, the rate of change of produced cost increases by $9/yd
State why you should use a table of values for the following: f(x) = 2^x. Estimate the value of f'(0).
1) find the derivative f'(0) = lim(h→0) [f(0+h) - f(0)]/h = lim(h→0) [2^h - 1]/h **can't simplify, plug into calculator find limit as h approaches 0 using table of values; f'(0) = 0.693
Let D(t) be the U.S. national debt at time t. The table below gives the approximate values of the function by providing end of year estimates, in billions of dollars, from 1980 to 2000. Interpret and estimate the value of D(1990) (1980,930.2) (1985, 1945.9) (1990, 3233.3) (1995, 4974.0) (2000, 5674.2)
1) locate year you're looking for (1990, 3233.3) 2a) when there are EQUAL INTERVALS of x (ex. 5 year separation) find rate of change ex. 1985 and 1995 2b) when there aren't equal intervals, find the average rate of change from the closest values to the left and right of the value and take the average of the two (1985, 1990) and (1990, 1995), then add the slopes and divide by 2
Consider a ball dropped from a height of 450 m. Find the velocity of the ball when it hits the ground.
1) solve for time 4.9t² = 450; t = 9.6 2) solve for speed v(9.6) = 94 m/s
How many decimal places should you use on the AP Calc exam when in doubt?
3 decimal places
Solve lim(x→∞) [5e⁻(x) + 10]/[6e^x - 1]
= 0 + 10/∞; C so 0
How would the derivative graph f'(x) look if f(x) is a semicircle with a turning point at x = 0, positive values on the right and negative on the left, and zeroes at x = -5 and 5?
A graph with a zero at x = 0, the right approaching a vertical asymptote y = -5 (positive) in an exponential manner, and the left approaching y = 5 (negative) in an exponential manner
Given a graph of f(x), which has a horizontal asymptote at 0, a turning point at (0,-4), and is negative as it approaches the y axis and from the left and positive as it moves away from the right, what graph would represent f'(x)?
A graph with turns at ±0.5, has its steepest section at x = 0 (origin), has a positive slope between -0.5 and 0.5, and is negative and approaching the origin outside of it
Given a graph of f(x), which has turning points at x = 3 and x = 5, with (-∞,3) having a negative slope, (3,5) having a positive slope, and (5,∞) having a negative slope, what graph represents f'(x)?
A graph with zeroes at 3 and 5 [(3,5) is above the x axis] and all values outside of that interval being negative
Given a graph of f(x), which has turning points or slopes of 0 at x = 0 and x = -1.5, with the slope between (-∞,-1.5) being negative, the slope between (-1.5,0) being positive, and the slope between (0,∞ being negative), what graph represents f'(x)?
A graph with zeroes at x = 0 and x = 1.5, with (-1.5,0) being positive and the sections outside of it being negative
If V(t) is the volume, in gallons, of water in a tank t hours after noon, and if V(2) = 200 and V'(2) = 5, then which of the following must be true? I) At 2 pm, the amount of water in the tank is decreasing at a rate of 5 gallons per hour II) At 3 pm, there will be fifteen gallons of water in the tank III) V(2) - V(0)/2 = -5
A), I only
If f is an increasing function whose graph lies below the x-axis and is concave up, then which of the following must be true? I) f(x) < 0 II) f'(x) < 0 III) f''(x) < 0
A), I only
Give examples of how you would state the values at which f is discontinuous on a graph: - x = -5 (removable discontinuity with point under) - x = -3 (asymptote) - x = 2 (right ≠ left)
At x = 5 because lim(x→-5) f(x) ≠ f(-5) At x = -3 because lim(x→-3) f(x) = DNE or f(-3) = DNE At x = 2 because lim(x→2) f(x) = DNE
If f(x) * f'(x) * f"(x) > 0, then which of the following is possible? a) the graph is in the second quadrant and is decreasing and concave up b) the graph is in the second quadrant and decreasing and concave down c) the graph is in the first quadrant and increasing and concave down d) the graph is in the third quadrant and increasing and concave up e) the graph is in the fourth quadrant and decreasing and concave down
B) second quadrant, decreasing, concave down
If lim(h→0) [f(4+h) - f(4)]/h does not exist, which of the following must be true? I) lim(x→4) f(x) does not exist II) f is not continuous at x = 4 III) f is not differentiable at x = 4
C, II and III only would have given derivative at x = 4; option 1 could equal any value and option 2 f(x) could be continuous and still not be differentiable
If f(p) = q means that, at a price of p dollars, a sandwich shop can sell q thousand sandwiches per week, then the equations f(4.5) = 1.2, f'(4.5) = -0.1 mean that: a) when the price of a sandwich is $4.50, the shop can sell 1200 sandwiches per week and the profit is decreasing at $100 per week b) the graph of f is a straight line going through (4.5,1.2) with slope -1 c) when the price of a sandwich is $1.20, the shop can sell 4500 sandwiches per week, but the shop loses $100 per week d) when the price of a sandwich is $4.50, the shop can sell 1200 sandwiches per week and the price is decreasing at ten cents per sandwich e) none of the above
E) none of the above at a price of $4.50, the shop can sell 1200 sandwiches per week and sells 100 less sandwiches per week per dollar
What represents the antiderivative of f?
F; F' = f
Given a graph of a function f(x) with the following slopes, how would you find the graph of its derivative f'(x)? A: x = 0, slope = 4 B: x = 1.3, slope = 1 C: x = 2, slope = 0 D: x = 3, slope = -1 E: x = 5, slope = -1 F: x = 6, slope = 0 **find slopes using mini tangent lines
Graph ordered pairs with (x, slope) zeroes at turns
What are dy/dx and df/dx examples of?
Leibowitz notation
You are given 3 graphs: f''(x) (a straight line), f'(x) (a parabola with two zeroes), and f(x) (a concave up and concave down combination). Using the points x1 (local maximum for f(x), x2 (local minimum for f(x), and x3 (steepest part of the negative slope between them), determine the relationship between these points across graphs and how the functions change.
X1: - at x1, f'(x) = 0 (turning point -> zeroes) - f' changes from positive to negative = local maximum X2: - at x2, f'(x) = 0 (turning point -> zeroes) - changes from - to + so local minimum X3: steepest point of area between a max and min = minimum for parabola (same a.o.s) - f"(x) = 0, changes signs from - to + (turning point -> zeroes)
What may f' have?
a derivative of its own
What do we often want to find instead of a function?
a function F whose derivative f is given
What is the limit as x→±∞?
a horizontal asymptote
You are given the graph y = f(x) which increases at (0,1), decreases from (1,3), and increases from (3,∞). Let F be the antiderivative of the function f graphed. a) Where is F increasing or decreasing? b) Where if F concave upwards or downwards? c) At what values of x does F have an inflection point? d) If F(0) = 1, then sketch the graph of F e) How many antiderivatives does F have?
a) increasing (0,∞), doesn't decrease b) CU: (0,1)u(3,∞); CD: (1,3) c) inflection point at x = 1, 3 d) concave up, concave down, concave up with point at (0,1) e) many antiderivatives (can slide graph up or down)
Let f(x) = {x² for x ≤ 3 and {Ax + B for x > 3 a) use the definition of the derivative to find A and B so that f'(3) exists. Justify your answer b) does f"(3) exist? Explain
a) lim(h→0⁻) [f(x+h)-f(x)]/h = 2x lim(h→0⁺) [f(x+h)-f(x)]/h = A for f'(x) to exist, 2x must equal A; at x = 3, 2*3 = 6 = A find B; if a function is differentiable it is continuous f(3) = (3)² = 9; lim(x→3⁻) = 9 and lim(x→3⁺) 6x + b = 18 + B; 9 = 18 + B or B = -9 b) at x = 3, f'(x) changes from a line with slope 2 to a horizontal line, creating a corner. Therefore, f'(x) is not differentiable at x = 3, or in other words f"(3) does not exist
How should the limits of piece-wise defined functions be determined?
algebraically (use function!!)
How can the instantaneous rate of change be interpreted?
as the slope of the tangent to the curve y = f(x) at P(x₁, f(x₁))
How would you find average velocity and initial velocity?
average: ∆s/∆t = lim(h→0) f(a + h) - f(a)/h initial: lim(h→0) f(a + h)- f(a)/h
concave
can be concave up or concave down; either + and - or - and positive
Solve lim(x→π/2) sin x/x
can insert value into equation; 1/(π/2) = 2/π
What is the derivative of sine?
cosine
Given a graph with a parabola, what would be the difference between f(2) = 3 and f'(2) = 3?
f(2) = 1, f'(2) = -1 (slope at 3)
How can you evaluate limits as x→c?
direct substitution
What should you do if you are given a graph and asked to consider the slopes of the curve at each point and list them from least to greatest?
draw tangent lines for each; negative slope < 0 < positive slope steepest of the negatives are the smallest, etc., steepest of positives are largest if really difficult use coordinate plane for rise/run
The position of a particle is given by the equation of motion s = f(t) = 1/(1+t), where t is measured in seconds and s in meters. Find the velocity and speed after 2 seconds. Solution: find the derivative of f when t = 2
f'(2) = [f(2+h) - f(2)]/h = [1/(3+h) - (1/3)]/h = (-h/(9+3h))/h * (9+3h/9+3h) = -h/h(9+3h) = -1/(9+3h) velocity = -1/9 m/s speed = 1/9 m/s
What is the derivative of a function f at number a, denoted by f'(a), represented by if the limit exists?
f'(a) = lim(h→0) [f(a+h) - f(a)]/h
How do you algebraically find DERIVATIVE functions?
f'(a) = lim(h→0) [f(a+h) - f(a)]/h **do not confuse with lim(x→c) f(x) which gives y coordinate
Find the derivative of the function f(x) = x² - 8x + 9 at at the number a. What is the derivative if a = 4?
f'(a) = lim(h→0) [f(a+h) - f(a)]/h lim(h→0) = [(a+h)² -8(a+h) + 9 - (a² - 8a + 9)]/h = h(h + 2a - 8)/h = h+ 2a - 8 sub in h → 0; f'(a) = 2a - 8 f'(4) = 2(4) - 8 = 0
For the function f(x) = x³ - x, use limits to find f'(x)
f'(x) = lim(h→0) [(x+h)³ - (x+h) -(x³-x)]/h f'(x) = lim(h→0) (x³ + 3x²h + 3xh² + h³ -x - h -x³ + x)/h f'(x) = lim(h→0) h² + 3x² + 3xh² - 1; sub in h = 0 f'(x) = lim(h→0) 3x² - 1
Describe a derivative as a function of x (replace a fixed number with a variable x)
f'(x) = lim(h→0) [f(x+h) - f(x)]/h
Describe the point of inflection for f'(x) and f"(x)
f'(x) = local max/min f"(x) = 0 and changes to sign
When is the rate of change increasing and when is it decreasing?
increasing when positive, decreasing when negative
How do you insert functions into a derivative (ex. if you press MATH and N(DERIVATIVE) on your calculator)?
input field: d/d[] () x = [] add in x; d/dx insert function by going to VARS, Y-VARS, FUNCTION, and Y1 x = x final: d/dx (y1) x = x
What are slopes reciprocals for?
inverse functions
What is d/dx and what does it tell you?
it is an operator; tells you to derive a function with respect to the variable x
What does it mean if something is differentiable at a point or interval?
it is continuous ex. differentiable at x = a implies continuous at x = a
Why is a derivative called a derivative?
it is derived from f(x) using limits
How does an asymptote translate to the graph of a derivative?
it stays in the same location but approaches the asymptote from either the top or bottom depending on direction (bc slopes are the same if approaching asymptote)
In example 3 we found the first derivative of f(x) = x³-x was f'(x) = 3x²-1. Repeat the process again to find f''(x).
lim(h→0) f''(x) = [f(x+h)-f(x)]/h [3(x+h)² - 1 -(3x² - 1)]/h (3x² + 6xh + 3h² -1 -3x² + 1)/h = h(3h + 6x)/h = 3h + 6x lim(h→0) 3h + 6x = lim(h→0) 6x
Using the definition of continuity at a point, explain if the function f(x) is continuous from the right or continuous from the left of two. Be very clear: f(x) = {3-x, x ≥ 1 {(x²-1)/(x+1), x < 1
lim(x→1⁻) f(x) = 2 lim(x→1⁺) f(x) = 2 continuous at x = 1 because lim(x→1) f(x) = f(1) = 2
Using the definition of continuity at a point, explain if the function f(x) is continuous from the right or continuous from the left of two. Be very clear: f(x) = {x²-4x+5, x < 2 {4 - x, x ≥ 2
lim(x→2⁻) f(x) = 1 lim(x→2⁺) f(x) = 2 continuous from right because lim(x→2⁺) f(x) = f(2) = 2
Find the following limit by showing all algebraic work: lim(x→∞) (6x + sin x)/x
lim(x→∞) 6x/x + lim(x→∞) sin x/x 6 + 0 = 6 0 because lim(x→∞) sin x/x = 0 (need to memorize)
Describe the equation for the instantaneous rate of change
lim(∆x → 0) = ∆y/∆x = lim(x₂ → x₁) [f(x₂)-f(x₁)]/x₂-x₁ **see example 4
Describe local maximums and minimums
local max: changes from + to - local min: changes from - to +
What should you always do when asked to interpret a graph?
look to see what type of graph it is
Find an equation of the tangent line y = x² - 8x + 9 at the point (3,-6)
m = f'(3) = f'(a) = 2a - 8 = 2(3) - 8 = 2
Find the equation of the tangent line to the parabola y = x² at the point (3,9) given the latter method
m = lim(h→0) [(3+h)² - 9]/h (9 + 6h + h² - 9)/h h(6 + h)/h m = lim(h→0) 6 + h; = 6 + 0 = 6 y - 9 = 6(x - 3)
Find an equation of the tangent line to the hyperbola y = 3/x at the point (3,1) using the latter formula
m = lim(h→0) [(3/(3+h) - 1]/h * (3+h)/(3+h) in numerator and denominator (3 - 3 - h)/h(3 + h) m = lim(h→0) 1/(3 + h) = 1/3+ 0 = 1/3 y - 1 = 1/3(x - 3)
What was the slope of the tangent to a curve with the equation y = f(x) and x = a in the preceding chapter?
m = lim(h→0) [f(a+h) - f(a)]/h
What is another formula for slope?
m = lim(h→0) [f(a+h) - f(a)]/h ** still ∆y/∆x x coordinate for first point on slope a , x for second a + h, distance is therefore h
If y = x² + 2x, given the point (-3,3) find the slope
m = lim(x→-3) (x² + 2x - 3)/(x - 3) = x - 1 = -4
Find the equation of the tangent line to the parabola y = x² at the point (3,9)
m = lim(x→3) (x²-9)/(x-3); remove hole = m = lim(x→3) x + 3 = 6 y - 9 = 6(x - 3)
When given a graph and asked to find other functions, what do you typically need?
more information/a point to locate where it is on the graph
Does the converse work (if f is continuous at a then f is differentiable at a)?
no; it can be a hole (ex. absolute value graph)
What is a vertical asymptote?
not just where the denominator is 0; it can also be a removable discontinuity
Describe the graph of y = f'(|x|)
sharp turn at 0 so no zero there, only removable discontinuities on either end of the function slope is -1 on left side and 1 on right side so straight lines with removable discontinuities on their respective sides at y = 1 (right) and y = -1 (left)
How should you prove a VA exists?
show lim(x→c) f(x) = ±∞; x = c = VA
How should you prove a HA exists?
show lim(x→±∞) f(x) = # y = # = HA
What is the difference between speed and velocity?
speed = absolute value (no direction) velocity = + or -, includes direction
What does the graph of f(x) = |x| capture?
that some functions have derivatives only at some values of x, not all
Describe the domains of f'(x) and f(x)
the domain of f'(x) may be smaller than the domain of f(x)
If the graph of f goes through the points (1,4), (2,6), and (3,10) then on the interval (1,3) the graph of f must be: a) increasing and concave up b) increasing and concave down c) decreasing and concave up d) decreasing and concave down e) none of the above
the graph could be increasing and decreasing and CU and CD; if you knew that f'(x) and f"(X) did not change signs you could make more conclusions
How do you find the limit as x→c? When is it DNE?
the left and right sides of the function must be equal; if they are not = DNE
What is the tangent line to the curve y = f(x) at the point P(a, f(a))?
the line through P with the slope lim(x→a) (f(x)-f(a))/(x-a) = m or ∆y/∆x
What happens at a local minimum?
the slope changes from positive to negative or negative to positive
What should you find if asked to find f'(a)?
the slope of the tangent line
velocity
the slope of the tangent line of the distance curve
Why can't you find the derivative at sharp turns?
there are no tan lines
In the preceding chapter, what was the formula for the velocity of an object with position function s = f(t) at t = a?
v(a) = lim(h→0) [f(a+h) - f(a)]/h
When is x the inflection point of f(x)?
when f"(x) = 0 AND f"(x) changes signs
What does this mean?
when x = 1000, it is increasing 9x as fast as x
What do inflection points tell?
where the concavity of the original f(x) function changes (uphill or downhill rests not min/max)
Consider a ball dropped from a height of 450 m. Find the velocity of the ball after 5 seconds using the latter method.
x = 5; use s(t) = 4.9t² (va) = lim(h→0) [4.9(a² + 2ah + h²) - 4.9a²]/h (9.8ah + 4.9h²)/h = 9.8a + 4.9h v(a) = 9.8a = 49 m/s