Exam 1- Functions, Limits, and Differentiation

Find where these two functions intersect: f(x)= 1/(1-x) and g(x)= 2x+1

(0,1) and (1/2, 2)

Find all critical points of the function f(x)= (x-1)/(x^2+3)

(3, 1/6) and (-1, -1/2)

For the function g(x)= 1/4x^4 - 4/3x^3, on what intervals is the function increasing?

(4, infinity)

Find the slope of the line tangent to the curve y=x^2root(1-x) at x=-3.

-1/3

What are the zeroes of g(x)= 2x^3-8x^2+6x

0,1,3

Differentiating Power Functions

1) Constant rule- If f(x)=a and a is a real number, then f'(x)= 0 2) Constant multiple rule- If f(x)=ax and a is a real number, then f'(x)=a 3) Power rule- If f(x)=x^n and n is a real number, then f'(x)=nx^(n-1)

Curve Sketching

1) Draw x and y axes and mark the domain on the x-axis. 2) Find the y-intercept and mark it on the y-axis. 3) Find f'(x), set equal to zero and solve for x. Your answers are the x-coordinates of the critical points. Mark these values on the x-axis. a) For each x value, find f(x) and then plot the points (x, f(x)). These are the critical points for the function. Since the slope is zero, mark the point with a small horizontal line. 4) Find f''(x) and calculate f''(x) at each of the x-values of the critical points. Apply the second derivative test to f''(x) to determine concavity. Mark a small up or down curve at each point to illustrate the concavity at each point. 5) Set f''(x) = 0. Determine if there are inflection points by determining the value of f''(x) on each side of the points found by solving. If the concavity changes on each side of x then there is an inflection point. 6) Using these values you can sketch a smooth curve.

Limit as X approaches A

1) Find functional value at x=a 2) If your answer is a number that value is the limit 3) If the value is an expression divided by zero then you must use algebraic techniques to find the limit- a) If f(a)= n/0 then the limit does not exist because we cannot divide by zero b) If f(a)= 0/0 then use algebra to simplify the expression and evaluate the limit

Steps to Solve Word Problems

1) Read the problem through once 2) Identify what you are being asked to find and underline 3) Draw a picture and label 4) Write each phrase as an equation- identify the equation that relates to the item you're solving for then solve other equations and substitute

Evaluate lim x->3 (x^2-4x+3)/(x^2-9)

1/3

You decide to build a box with a square base and with a height equal to half the length of a bottom side. The material you choose for the top of the box costs $6 per square foot. Since everyone will see the top of the box, that material is more expensive than what you choose for the sides of the box and the base: $4 per square foot. What equation expresses the cost of your box in terms of the height of the box? Be sure to show a labeled picture of the box.

10x^2+8hx

A frozen yogurt stand makes a profit of P(x)= .4x-80 dollars when selling x scoops of yogurt per day. How many more scoops of yogurt will need to be sold to raise the daily profit from $30 to $40?

25 more scoops

Evaluate lim x->infinity (3x^2-4)/(x^2+1)

3

Derivative

A derivative of a function represents the rate of change. Differentiation is the process of calculating a derivative.

You have been asked to build a rectangular garden for your neighborhood. The garden is to have an area of 720 square feet. The garden must have an 8 foot wide walkway on the north and south sides and a 10 foot walkway on the east and west sides. Find the equation for the total area of the project (garden and walkways). Be sure to show a labeled picture.

A= (x+20)(y+16)

Graphing a Function

Find y-intercept (when x=0) then x-intercepts (when y=0). X-intercepts are the zeros.

Rate of Change

How a function is changing as the values of x change from x=a to x=b

Find Equation of Line That Goes Through Two Points

If a line goes through (x1, y1) and (x2, y2) then the slope is given by: m= (y2-y1)/(x2-x1) and put into equation y-y1= m(x-x1)

Using the techniques of calculus (first and second derivative for local max/min and concavity) graph the function f(x)= -1/3x^3+2x^2-11.

Min- (0,-11) Max- (4,-1/3) Inflection point- (2,-17/3) In this general shape

An average sale at a small toy shop is $28, so the weekly revenue function is R(x)= 21x , where x is the number of sales made in 1 week. The weekly cost is given by the function C(x)= 7.2x+800 dollars. What function can the owner use to calculate profit per week?

P(x)= 13.8x-800

Differentiating Products

Product rule: (f(x)*g(x))'= f'(x)g(x) + f(x)g'(x)

Differentiating Quotients

Quotient rule: (f(x)/g(x))'= (f'(x)g(x)-f(x)g'(x))/(g(x))^2

Domain

The set of all values which can be put into the function. Usually all real numbers except is undefined with fractions divided by zero and negative square roots.

Slope of the Tangent Line

The slope of the tangent line at x=a represents increase or decrease for the tangent line and tells us if the rate of change for f(x) at x=a is increasing or decreasing

For this function, f(x)= (x^3/3)-2x+4pie, find all values of x where the slope of the tangent line is 16.

X= 3root2 and -3root2

Determine the equation of the tangent line to the curve f(x)= x^3-4x^2+6 at x=2.

Y= -4x+6

For the function f(x)= x^3-6x^2-15x-1: a) Find any inflection points b) On what intervals is the function concave up? c) On what intervals is the function concave down?

a) (2,-47) b) (2,infinity) c) (-infinity,2)

The number of minutes it takes a rat to run a maze is given by the function T(n)=4+(18/n+1) where n is the number of times the rat has run the maze. a. What is the average rate of change from 1 to 2 trials? b. What is the instantaneous rate of change when n = 2? c. Explain what happens to the time as the rat runs more and more mazes?

a) -3 mins b) -2 mins c) As the rat runs mores, the time it takes to run a maze decreases

A toy rocket is placed on the ground and fired straight into the air. The height of the rocket is given by h(t)= 64t-16t^2 where t is measured in seconds. Find: a) The velocity when t = 2. b) The acceleration when t = 2. c) What is the shape of the height function?

a) 0 b) -32 c) It is a parabola

Concavity

a) A function is concave up at x=a if the curve of the function always lies above the tangent line (the graph could hold water) b) A function is concave down at x=a if the curve of the function always lies below the tangent line (the graph would spill water) c) Inflection point= If f'(a)=0 and f(x) changes concavity at x=a, the x=a is an inflection point

Absolute Minimum and Maximum

a) Absolute maximum is the largest value the function takes on its domain b) Absolute minimum is the smallest value the function takes on its domain

Limit as X approaches Infinity

a) If f(infinity)= n/infinity the limit is zero b) If f(infinity)= infinity/infinity then divide all terms in numerator and denominator by the largest term in the denominator c) If f(infinity)= infinity there is no limit

Increasing/Decreasing Function

a) If f(x) gets larger as x gets larger, then f is increasing b) If f(x) gets smaller as x gets larger, then f is decreasing c) A function may exhibit both increasing and decreasing patterns d) We describe where a function is increasing or decreasing in terms of x-intervals

Relative Minimum and Maximum

a) If f(x) is increasing and then changes to decreasing at x=a, f(x) has a relative maximum at x=a. The relative max is at point (a, f(a)) and f(a) is the max value. b) If f(x) is decreasing and then changes to increasing at x=a, x=a is a relative minimum at point (a, f(a)) and f(a) is the min value.

Instantaneous Rate of Change

a) The derivative is the rate of change at a particular point in time (dy/dx) b) To find the instantaneous rate of change, find the derivative. Then evaluate at a point and estimate the value.

Let f(x)= 2-x^2 a. Find f(a+2) b. Find f(x+1)-f(x)

a. -a^2-4a-2 b. -2x+1-2

Higher Order Derivatives

f'(x), f''(x), f'''(x), etc. Velocity is the first derivative of the position function, acceleration is the second derivative of the position function, and marginal cost is the first derivative of a cost function.