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The final exam for a college-level mathematics course, math 105. The exam covers various topics including derivatives, integrals, and calculus applications. Students are required to show all their work for partial credit. Questions include computing derivatives, finding antiderivatives, estimating integrals, sketching graphs, and minimizing costs.
Typology: Exams
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While a final answer is important, you earn points for all the work leading to that answer, not just the answer itself. Show all your steps clearly so you will be eligible for the most partial credit. Good luck!
1.) Compute the following derivatives:
a.) (5 pts.)
d dx
∫ (^0). 7 x
5
√ arcsin(t^3 ) dt
b.) (5 pts.)
dy dx
, where y =
eax^ − e−ax eax^ + e−ax
2.) (10 pts.) An object thrown in the air on a planet in a distant galaxy is at height s(t) = − 12 t^2 + 83t + 62 feet at time t seconds after it is thrown. What is the acceleration due to gravity on this planet? With what velocity was the object thrown? From what height?
3.) Find both left hand (5 pts.) and right hand (5 pts.) estimates for
∫ (^12) 0
f (x) dx, with
∆x = 2, using the table below.
x 0 2 4 6 8 10 12 f (x) 1 3 7 12 8 4 15
5.) Given the graph of f ′(x) below:
a.) (5 pts.) sketch its derivative f ′′(x) on the first set of blank axes, and
b.) (5 pts.) use both f ′(x) and f ′′(x) to sketch the graph of f (x) on the last set of axes.
6.) (10 pts.) A builder intends to construct a storage shed having a volume of 900 ft^3 , a flat roof, and a rectangular base whose width is three-fourths the length. The cost per square foot of the materials is $4 for the floor, $6 for the sides, and $3 for the roof. What dimensions will minimize the cost?
a.) (5 pts.) Does the function graphed above appear to satisfy the hypotheses of the Mean Value Theorem? Explain your answer fully.
b.) (5 pts.) Does it satisfy the conclusion? Again, explain fully.
NOTE: be sure to make clear what the hypotheses and conclusion of the Mean Value Theorem are.
10.) Using the information in the table about f and g, find
a.) (2 pts.) h(4) if h(x) = f (g(x))
b.) (2 pts.) h′(4) if h(x) = f (g(x))
c.) (2 pts.) h′(4) if h(x) = g(f (x))
d.) (2 pts.) h′(4) if h(x) = f (x)/g(x)
e.) (2 pts.) h′(4) if h(x) = f (x) · g(x)
x 1 2 3 4 f (x) 3 2 1 4 f ′(x) 1 4 2 3 g(x) 2 1 4 3 g′(x) 4 2 3 1
