APPLIED CALCUS EXAM SOLUTIONS

Exam 2

Math 261: Applied Calculus I

April 21, 2020

Concepts. (3 points each)

Answers here should be as simple and non-technical as you can make them.

1. Explain the difference between a relative and absolute maximum.

2. The first derivative of a function, roughly speaking, measures how fast a function

is changing at a point x. That is, for a function f and point x, f′(x) tells us how

fast the function is changing at the point x. What does f′′(x) tell us? (Hint:

Think of the first derivative as the speed of a car, and the second derivative as

the car’s acceleration.)

3. (True/False) If a function f has a critical point at x, then f(x) must either be a

relative maximum or relative minimum.

4. (True/False) An inflection point of a function is a point across which the direction

of concavity changes.

5. (True/False) If a function is increasing on an interval I, then f′(x) > 0 at every

point x on the inverval I.

6. (True/False) If a function is concave down on an interval I, then f′(x) < 0 at

every point x on the inverval I.

1

2

Computation

1. For the following function, list all the critical points and state whether or not the

critical point is a relative maximum or minimum (8 points):

x1: relative min

2. Sketch the graph of f(x) = x3 − 3x + 6 by following the procedure below. In each part you must show all your work for full credit.

Find f′(x) (5 points):

Find the critical points of f. State them in the form (x, y). That is, for each

critical point, find the corresponding y value and write the critical point as an

ordered pair (5 points):

Determine whether each critical point c is a relative maximum, relative minimum,

or neither (5 points):

3

Sketch the graph of f (showing concavity and inflection points is not necessary,

but you must make clear intervals where f(x) is increasing/decreasing and what

f(x) looks like at each critical point) (5 points):

3. Sketch a graph that matches the following description (4 points):

G(x) has a negative first derivative over (−∞,−3) and a positive first derivative over (−3,∞)

4. Sketch a graph that matches the following description (4 points):

G(x) has a positive first derivative over (−∞,−5), a negative first derivative over (−5, 0), and negative first but positive second derivatives over (0,∞).

4

5. Suppose f(x) = x3 −12x. Sketch the function by following the procedure below. In each part you must show all your work for full credit.

Find f′(x) and f′′(x) (6 points):

Find the critical points and classify each as producing a relative max, relative

min, or neither. As before state each point in the form (x, y) (6 points):

Find the inflection points. Make sure you show your justification for why the

point is an inflection point (6 points):

On which intervals is the function increasing, and which is it decreasing? (6

points):

On which intervals is the function concave up, and which is it concave down?

(6 points):

5

Sketch the graph (you must show what the function looks like at each criti-

cal point, inflection point, and on the intervals you listed above) (6 points):

6. Sketch a graph that matches the following description (4 points):

f is increasing and concave up on (−∞, 0), f is increasing and concave down on (0,∞).

7. Sketch a graph that matches the following description (6 points):

f(0) = 5, f′(0) = 0, f′′(0) < 0

f(2) = 2, f′(2) = 0, f′′(2) > 0

f(4) = 3, f′(4) = 0, f′′(4) < 0