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Mathematics 105 — Calculus I
Exam 1
October 3, 2008
Your Name:
There are 6 problems in this exam. On each problem, you must show all your work, or otherwise thoroughly explain your conclusions. There is always at least one step preceding a final answer. Units may be requested for your final answer; a point deduction will apply if they are omitted.
On the portion of the exam marked N C, you will be allowed 20 minutes during which your calculator must be closed and put away. If you finish this section early, you may hand in your work early. However, only after you hand in the ”no calculators” section will you be permitted to use a calculator.
You will have 55 minutes to complete this exam.
Question Point Value Your Score
No Calc. 50
Total 150
N C P
Math 105-D (Salomone) Exam 1 Show all your work!
Name:
Score (50 possible):
Problem 1-NC. (20 points) Consider the function h defined by the formula
h(x) =
x^4 +
x x
(a) (10 points) What is the natural domain of h? Why?
(b) (10 points) Find an antiderivative H which satisfies H(1) = 14.
Hint: You might want to perform some algebra before doing any calculus.
Math 105 Exam 1 Score this page: Problem 1. (20 points) An electrocardiogram (EKG) is a record of the amplitude A of electrical activity in the heart as a function of time t. Shown below is an EKG of a (mostly) healthy human heart.
A
t
P QR S T
(a) (5 points) Which of the following words describes the func- tion A? Circle all that apply.
periodic even odd continuous
polynomial exponential logarithmic constant
(b) (5 points) Compare and contrast the slope of A(t) between points Q and R and the slope between points R and S.
(c) (5 points) Circle one:
- A′(P) is... negative zero positive
- A′′(P) is... negative zero positive
What do your answers imply about the function A at the point P?
(d) (5 points) Explain what A′(R) is, using the word ”limit” in your answer.
Math 105 Exam 1 Score this page: Problem 2. (25 points) Inside a mystery box is a differentiable function f whose derivative f ′^ is graphed below.
(a) (6 points) On what interval(s) is f an increasing function?
(b) (6 points) On what interval(s) is f concave up? 5
f’
−2 −1 1 2 3 4 6
y
x
(c) (6 points) What kind of stationary point does f have at x = −2? Why?
(d) (7 points) Using the axes provided, sketch a graph of the function f. 6
f
−2 −1 1 2 3 4 5
y
x
Math 105 Exam 1 Score this page: Problem 4. (30 points) Because free fall just isn’t enough for some people, and terminal velocity is ”so last century,” the newest craze in extreme sports is jetpack skydiving, where a jetpack is used to accelerate the skydiver toward the ground.
On her most recent jetpack skydive, Eve recorded her velocity during the dive to impress her friends. Upon arriving home, she fit the polynomial function
v(t) = 66 t − 11 t^2
to her velocity, where v is measured in meters/second and t is in minutes.
(a) (15 points) Use a difference quotient with h = 0 .01 to estimate v′(1), her acceleration after 1 minute. Include units in your answer.
(b) (15 points) By finding her distance function d(t), determine how many meters Eve travelled during the first 3 minutes of her jetpack skydive.
Hint: You may assume d(0) = 0.
