AB Calculus - Hardtke Assignment: Derivatives D Name _______________________________________

Due: Thursday, 10/4 4pts

1. A) Sketch a curve whose slope is always positive and increasing.

B) Sketch a curve whose slope is always positive and decreasing.

C) For each of these descriptions, give an equation of such a curve.

2. A graph of a population of yeast cells in a new laboratory culture as a function

of time is shown.

A) Describe how the rate of population increase varies.

B) When is the rate highest?

C) On what intervals is the population function concave upward or downward?

D) Estimate the coordinates of the inflection point.

3. A particle is moving along a horizontal straight line. The graph of its position function

(the distance to the right of a fixed point as a function of time) is shown.

A) When is the particle moving toward the right and when is it moving toward the left? B) When does the particle have positive acceleration and when does it have negative

acceleration?

4. Use the given graph of f to estimate the intervals on which the derivative f ‘ is increasing or decreasing.

5. Coffee is being poured into the mug shown in the figure at a constant rate (measured in volume per

unit time). Sketch a rough graph of the depth of the coffee in the mug as a function of time.

Account for the shape of the graph in terms of concavity. What is the significance of the

inflection point?

Over

For 6 & 7, the graph of the derivative f ‘ of a continuous function f is shown. Answer each of the following questions:

A) On what intervals is f increasing? Decreasing?

B) At what values of x does f have a local maximum? Local minimum?

C) On what intervals is f concave upward? Concave downward?

D) State the x-coordinate(s) of the point(s) of inflection.

E) Assuming that f(0) = 0, sketch a graph of function f.

6.

7.

For 8 -10, sketch the graph of a function that satisfies all of the given conditions.

8. The first derivative is always negative, but the second

derivative is always positive.

9. f ‘ (0) = f ‘ (4) = 0, f ‘ (x) > 0 if x < 0.

f ‘ (x) < 0 if 0 < x < 4 or if x > 4.

f’’(x) > 0 if 2 < x < 4. f’’(x) < 0 if x < 2 or x > 4.

10. Let f(x) = x3 – x. Find: A) the intervals on which f is increasing or decreasing; B) the intervals on which f is

concave upward or downward; C) the inflection point of f. Show work that present a mathematical argument.

AB Calculus - Hardtke Assignment: Derivatives D SOLUTION KEY

Due: Thursday, 10/4 4pts

1. A) Sketch a curve whose slope is always positive and increasing.

B) Sketch a curve whose slope is always positive and decreasing.

C) For each of these descriptions, give an equation of such a curve.

2. A graph of a population of yeast cells in a new laboratory culture as a function

of time is shown.

A) Describe how the rate of population increase varies.

B) When is the rate highest?

C) On what intervals is the population function concave upward or downward?

D) Estimate the coordinates of the inflection point.

3. A particle is moving along a horizontal straight line. The graph of its position function

(the distance to the right of a fixed point as a function of time) is shown.

A) When is the particle moving toward the right and when is it moving toward the left? B) When does the particle have positive acceleration and when does it have negative

acceleration?

4. Use the given graph of f to estimate the intervals on which the derivative f ‘ is increasing or decreasing.

5. Coffee is being poured into the mug shown in the figure at a constant rate (measured in volume per

unit time). Sketch a rough graph of the depth of the coffee in the mug as a function of time.

Account for the shape of the graph in terms of concavity. What is the significance of the

inflection point?

Over

For 6 & 7, the graph of the derivative f ‘ of a continuous function f is shown. Answer each of the following questions:

A) On what intervals is f increasing? Decreasing?

B) At what values of x does f have a local maximum? Local minimum?

C) On what intervals is f concave upward? Concave downward?

D) State the x-coordinate(s) of the point(s) of inflection.

E) Assuming that f(0) = 0, sketch a graph of function f.

6.

7.

For 8 -10, sketch the graph of a function that satisfies all of the given conditions.

8. The first derivative is always negative, but the second

derivative is always positive.

9. f ‘ (0) = f ‘ (4) = 0, f ‘ (x) > 0 if x < 0.

f ‘ (x) < 0 if 0 < x < 4 or if x > 4.

f’’(x) > 0 if 2 < x < 4. f’’(x) < 0 if x < 2 or x > 4.

10. Let f(x) = x3 – x. Find: A) the intervals on which f is increasing or decreasing; B) the intervals on which f is

concave upward or downward; C) the inflection point of f. Show work that present a mathematical argument.