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Mathematics 105 — Calculus I
Exam 1
February 13, 2009
Your Name:
There are 6 total 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 30 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 may not return to the ”no calculator” portion after handing it in.
Before beginning, ensure your calculator is set to Radians mode.
You will have 80 minutes to complete this exam.
Question Point Value Your Score
No Calc. 50
Total 150
N C P
Math 105-A (Salomone) Exam 1 Show all your work!
Name:
Score (50 possible):
Problem 1-NC. (25 points) Use the limit definition of derivative to compute f ′(x) for the function
f (x) = x ex.
Hint: Simplify using properties of exponentials. You will also need to know that lim h→ 0 e
h (^) − 1 h =^ 1.
Math 105 Exam 1 Score this page: Problem 1. (25 points) This problem concerns the function
g(t) =
(t − 3)(t^2 − t + 1) t^2 − 4 t + 3
(a) (8 points) Determine the domain of this function.
(b) (10 points) Using algebra, compute lim t→a g(t) for each value of a not in the domain of g. Explain what each result
means about the continuity of g.
1 3 4
4 3 2 1
2
t
y = g(t)
(c) (7 points) At left is a partial graph of g(t). Fill in the gap, clearly indicating the nature of any discontinuities.
Math 105 Exam 1 Score this page:
Problem 2. (25 points) A weight is attached to a spring and suspended in a container of motor oil. If it is allowed to oscillate, its vertical position (measured in cm above equilibrium) as a function of time t in seconds might be given by the function
p(t) = 3 e−t^ cos t.
(a) (10 points) Complete the data table below, and use your results to estimate the values of p′(0.9), p′(1), and p′(1.1). Include units in your answers.
t (sec) 0.85 0.9 0.95 1 1.05 1.1 1.
p (cm)
p′(0.9) ≈ p′(1) ≈ p′(1.1) ≈
(b) (10 points) Use your answers to part (a) to estimate p′′(1), with units. What does this answer mean in practical terms?
(c) (5 points) Is it reasonable, based on your answers, to expect that p(t) satisfies the differential equation
p′′^ + 2 p′^ = − 2 p?
Why or why not?
Math 105 Exam 1 Score this page:
Problem 4. (25 points) The latest press booklet for the 2009 Lotus Exige S-240 sports car claims it can accelerate from 0—60 mph in 4.0 seconds flat. A recent test-track run showed that under full throttle, the velocity of the car is modeled by the function
v(t) = 40
t − 5 t,
where v is measured in mph and t in seconds.
(a) (15 points) According to this model, after how many seconds will the car reach its maximum velocity, and what is the maximum velocity?
Note: do this symbolically, showing your work. You may include a graph or data table if you wish, but your answer must be exact.
(b) (10 points) Determine the car’s distance function d(t) — an antiderivative of its velocity — and use it to find the distance the car traveled during the first 10 seconds of this time trial.
Note: write out the units of the antiderivative in your answer. Convert them if you wish.
