Document Title - Portland Community College
MTH 251
Final Exam – No Calculator Name
For each non-“fill in the blank” question, you must show all relevant work in a well documented manner to earn full credit for the problem.
1. Find the equation of the tangent line to the curve [pic] at the point where [pic].
(5 points)
2. On what interval(s) is the function [pic] decreasing? How do you know? (5 points)
3. Find the first derivative formula for each function. Make sure that you use proper notation on both sides of the equal sign. (2 points each)
a. [pic] b. [pic]
c. [pic] d. [pic]
e. [pic] f. [pic]
4. Choose the appropriate word or phrase from Table 1 that makes each given statement true.
(1 point each)
Table 1: Possible correct answers to problem 5.
|positive |negative |positive, zero, or undefined |negative, zero, or undefined |
|zero or undefined |zero |increasing |decreasing |
|concave up |concave down |horizontal |vertical |
a. At any point over intervals where [pic] is decreasing, [pic] is definitely (blank a).
b. At any point that an antiderivative of [pic] has a local maximum point, then it’s just got to be the case that [pic] is (blank b).
c. At any point that the slope of [pic] is positive, there’s no doubt that [pic] is (blank c).
d. Along intervals where [pic] is (blank d), we can be certain that [pic] is concave down.
e. If [pic] is everywhere differentiable, and [pic] has a point of inflection at [pic], then without a doubt the tangent line to [pic] at [pic] is (blank e).
f. It’s a true fact that [pic] is (blank f) at any point that [pic] is increasing.
The blanks
a.
b.
c.
d.
e.
f.
5. Find the slope of the tangent line to the curve with equation [pic] at the point [pic]. Make sure that your conclusion is fully substantiated.
(8 points)
6. On what interval(s) is the function [pic] concave up? How do you know?
(4 points)
7. Find the vertical asymptote(s) and horizontal asymptote(s) of the function [pic]? (4 points)
8. Find [pic]. (4 points)
9. Fill in each of the blanks with the value of each limit. (4 points)
a. [pic]
b. [pic]
c. [pic]
d. [pic]
10. Find the critical numbers of the function [pic]. Show all relevant work. Remember that your formula for [pic] needs to be a single, completely factored, fraction! (8 points)
11. For the function [pic], you are given that [pic].
a. What are the critical numbers of [pic]? No explanation necessary! (3 points)
b. Build an increasing/decreasing table for [pic]. (3 points)
c. What are the local minimum and local maximum points on [pic]? No explanation necessary! (3 points)
12. Several values related to a function of unknown formula are given in Table 2. Find each of the following in relation to this function.
(6 points total)
a. Find [pic] if [pic].
b. Find [pic] if [pic].
13. Find the first derivative formula for the function [pic]. (5 points)
14. Find the first derivative formula for the function [pic]. (5 points)
15. The side labeled [pic] in Figure 2 is changing at a constant rate of [pic]. Find the rate at which the side labeled [pic] is changing at the instant when [pic]. Write your conclusion using a complete sentence. (3 points)
16. True or False questions. Please answer each question below with regard to the function shown in Figure 3. Circle T or F (1 point each)
T or F The given function is decreasing at [pic].
T or F The first derivative of the given function is decreasing at [pic].
T or F An antiderivative of the given function is decreasing at [pic].
T or F The given function is nondifferentiable at [pic].
T or F The first derivative of the given function is nondifferentiable at [pic].
T or F An antiderivative of the given function is nondifferentiable at [pic].
T or F The tangent line at [pic] to an antiderivative of the given function is horizontal.
T or F If the given function is [pic], then [pic].
T or F If the given function is [pic], then [pic].
T or F If the given function is [pic], then [pic].
T or F If the given function is [pic], then [pic].
T or F If the given function is [pic], then [pic].
-----------------------
x
2 ft
y
Figure 2: [pic] and [pic] are lengths (in feet)
Figure 3: Function for problem 16
y
x
Table 2: Values for Problem 12
|[pic] |[pic] |[pic] |
|0 |-ð1 |3 |
|1 |-ð1 |-ð2 |
|2 |-ð3 |-ð1 |
|3 |-ð1 |6 |
411 lengths (in feet)
Figure 3: Function for problem 16
y
x
Table 2: Values for Problem 12
[pic][pic][pic]0−131−1−22−3−13−16
|4 |11 |19 |
|5 |39 |38 |
|6 |89 |63 |
Figure 1: [pic]
y
x
3 ft
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