M 160 Exam 2 Study Guide - Open Computing Facility



1. (10 points) The function g(x) and its derivative g'(x) have the values shown in the following table:

[pic]

The function f(x) is given by the expression f(x) = [pic] .

(a) P(x) = f(x) g(x) Find P′(0). (And show how you did it!)

(b) Q(x) = [pic] Find Q′(1). (And show how you did it!)

(c) F(x) = fog(x) Find F′(–2). (And show how you did it!)

2. (10 points)

(a) Use calculus and algebra (not calculator or graph) to find the exact values of the critical points of the function f(x) = [pic] on the interval [-1, 4]. Show clearly how you found these critical points.

(b) Use information from (a) to determine the interval(s) where the function given in (a) is decreasing. Your solution must use calculus, not the graph, and must be explained fully.

(c) Sketch a graph of the function on the coordinate system provided. (You may use your calculator.)

What do you see in the graph that indicates you probably found the critical points correctly?

3 –

2 –

1 –

| | | | | | x

-2 -1 0 1 2 3 4

-1 –

-2 –

-3 –

–

–

3. (10 points) 2–

(a) Sketch an accurate graph of the function

F(x) = [pic] 1–

-2 -1 0 1 2

(b) Do you believe the function in (a) is differentiable at the point x = 1?      YES      NO

Describe what you see in the graph that leads you to this conclusion.

Explain how what you see in the graph supports your conclusion.

(c) Use the definition of the derivative (as the limit of a difference quotient) to show that your conclusion in (b) is correct.

4. (10 points) A water storage tank is in the shape of a circular cone with point up as shown in the figure. The bottom of the tank is a circle with diameter 8 feet and the top of the tank is the point of the cone. The tank is 10 feet tall. Water is being pumped into the tank at the rate of 5 ft3/min. How fast is the water level rising when the water level is 6 feet from the top of the tank?

[pic]

5. (10 points) (a) Show that the point (1, 2 ) is on the graph of 4x2 + 4y2 = [pic].

(b) The graph of 4x2 + 4y2 = [pic] is shown below. Explain why this is not the graph of a function.

Then circle or otherwise identify a portion of the graph of 4x2 + 4y2 = [pic] that defines a function y = f(x) that could be differentiated to find the slope of the line tangent to this curve at the point ( 1, 2).

[pic]

(c) We don’t have an explicit equation for the function y = f(x) you defined graphically in part (b).

Find an expression for [pic] = f′(x) in terms of x and y = f(x). Show the details of your calculations.

(d) Write an equation in point-slope form for the line tangent to the curve at the point ( 1, 2).

(e) The point (1, ½ ) also lies on the graph of 4x2 + 4y2 = [pic]. Could you use the formula for f′(x) you derived in (c) to find the slope of the tangent line at (1, ½ ) even though this point is not on the graph of

y = f(x)? Explain why or why not.

( “Because that’s the way implicit differentiation works” isn’t a satisfactory explanation.)

6. (10 points) One definition of the derivative of a function

f at a number c is f'(c) =[pic][pic].

The figure shows the graph of a function y = f(x) and

a number c marked on the x-axis.

(a) Illustrate (draw!)and label each of the quantities

h, c + h, f(c), and f(c+h) that appear in this definition on the graph. (You may assume h > 0.)

| x

c

(b) Illustrate (draw!) and explain how to interpret the number f'(c) in terms of the graph.

(c) Illustrate (draw!) and explain how to interpret the expression [pic] in terms of the graph.

(d) Explain in non-technical terms what the symbol [pic] means in the equation

f'(c) =[pic][pic] that defines the derivative of a function f at a number c .

7. (10 points) (a) Show that x = 0 is a critical point of the function f(x) = [pic] .

(b) If possible, sketch the graph of a function that has all these properties:

(i) the function is defined and continuous of the interval [-1, 4]

(ii) x = 0 is a critical point of the function;

(iii) the function does not have an extreme value at x = 0.

If it is impossible to give such an example, explain how you know.

3 –

2 –

1 –

| | | | | | x

-2 -1 0 1 2 3 4

(c) Write a complete statement of a theorem that ensures that any function that meets the requirements in (b) has an absolute extreme value at some point in the interval [-1, 4].

(d) Use calculus and algebra to find the absolute maximum and absolute minimum of the function

f(x) = [pic] on the interval [-1, 4]. At what point(s) does the function attain these values?

8. (30 points) Calculate the indicated derivatives. Simplification is not required. If you simplify, you must do it correctly.

(a) y = [pic] [pic]

[pic]

(b) g(t) = (4 + t – 5t2) cos(t) g'(t) =

(c) f(x) = sec[pic] f'(x) =

(d) q(x) = [pic] q'(x) =

(e) C(x) = [pic] C'(x) =

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