First exam practice sheet



Dates of test Wednesday, Nov 10 to Monday, Nov 15 in testing center

Material covered Ch 1 (sections 1 through 6)

Ch 2 (all)

Ch 3 (all)

Ch 4 sections 2 through 5)

Allowable materials Calculator (no TI-89 or 92)

You should know how to find all derivatives we have covered, including all trigonometric functions, natural log, and the inverse sine and inverse tangent functions.

Sample problems

1. Find the derivatives

a. [pic]

b. [pic] if [pic]

c. [pic]

d. [pic]

e. [pic] if [pic]

2. Use implicit differentiation to find an equation of the tangent line to the curve [pic] at the point [pic]

3. [pic]

4. Over what interval is the function [pic] concave downward?

5. Use logarithmic differentiation to find:

a. [pic]

b. [pic]

6. If [pic], find [pic]

|[pic] |-2 |-1 |0 |1 |2 |

|[pic] |3.4 |2.2 |1.4 |0.8 |0.6 |

|[pic] |-2 |-1 |0 |1 |2 |

|[pic] |-2.6 |-2 |-1.6 |-1.4 |-1.3 |

7. Determine which of the families of curves are the orthogonal trajectories to the family of curves [pic] and prove that they are orthogonal.

a. [pic]

b. [pic]

8. For the function [pic]

a. Find the linearization [pic] at [pic]

b. Determine the values of [pic] for which the approximation is within 0.1

9. The graph of [pic] is drawn below. Find the x-values at which the following occur:

[pic]

a. Local extrema of [pic]

b. Absolute extrema of [pic]. (guess and explain your choice).

c. Inflection points of [pic]

10.

11. Find the absolute maximum and minimum of the function [pic] over the interval [pic]

12. For the function [pic], over the interval [pic]

a. At what x-values does [pic]have a local maximum?

b. At what x-values does [pic] have a local minimum?

13. Evaluate the following limits. Use L’Hospital’s Rule if it applies.

a. [pic]

b. [pic]

c. [pic]

d. [pic]

e. [pic]

f. [pic]

g. [pic]

h. [pic]

14. Find the relative extrema, if any, of the function [pic]

15. Find the relative extrema, if any, of the piecewise function [pic]

16. Without finding the absolute extrema, determine which of the functions has an absolute maximum and and absolute minimum on the interval [pic]. If not, explain why the criteria for the Extreme Value Theorem are not satisfied.

a. [pic]

b. [pic]

c. [pic]

17. For the function whose graph is drawn below, identify the x-values in the open interval [pic] where the function:

18. [pic]

a. is not differentiable

b. has a relative minimum

c. has a relative maximum

19. Find all of the following for the function [pic]

a. Local extrema, if any

b. Horizontal asymptote, if any

c. Vertical asymptotes, if any

d. Inflection points, if any

e. Draw a graph of [pic] that captures all of the above aspects of the function. (You may use a graphing calculator).

20. For the polynomial function [pic]

a. Use a graphing calculator and derivatives to estimate the local extrema

b. Select a viewing window that shows all turning points.

c. Graph the polynomial in this viewing window.

Answers:

1.

a. [pic]

b. [pic]

c. [pic]

d. [pic]

e. [pic]

2. [pic]

[pic]

3. [pic]

4.

a. [pic]

b. [pic]

5. –0.6

6. a is the family of orthogonal trajectories

7.

a. [pic]

b. [pic]

8.

a. local max at [pic]

local min at [pic]

b. absolute min at [pic]

absolute max at [pic]

(decrease from –3 to 2 is greater than increase from –4 to –3 or from 2 to 3)

c. Inflection points at the following

[pic]-values: [pic]

9. absolute max = 1 at [pic]

absolute min = -9.392 at [pic]

10.

a. [pic]

b. [pic]

11.

a. [pic]

b. [pic]

c. [pic]

d. [pic]

e. [pic]

f. [pic]

g. [pic]

h. [pic]

12. relative min of 12 at x=2

13. relative max of 1 at x = 0

relative min of 0 at x = 1

relative min of 0 at x = 2

or anywhere on interval [pic]

14.

a. no, asymptote at [pic] so [pic] is not defined at [pic]

b. yes

c. no, asymptote at [pic], so [pic] is not continuous over the interval [pic]

15.

a. not differentiable at [pic]

b. relative min at [pic]

c. relative max at [pic]

16.

a. local min: [pic]

b. horizontal asymptote: [pic]

c. vertical asymptote at: [pic]

d. inflection point: [pic]

e. viewing window: [pic]

[pic]

17.

a. Extrema: [pic]

[pic]

b. Window: [pic]

(or similar)

c.

[pic]

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