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Main points of this exam paper are: Fundamental Theorem, Definite Integral, Calculus, Graph, Function, Sketch, Table, Estimate, Square Bottomed Box, Fixed Volume
Typology: Exams
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MATH 105 FINAL EXAM April 10, 2003, 8:00a.m.
Name:
Please show all steps clearly: your grade comes from your process and explanation, rather than just a final answer. Good luck!
1.) (10 pts.) Compute the definite integral, using the Fundamental Theorem of Calculus.
∫ (^2) π
0
(sin x + cos x) dx
2.) (10 pts.) Given the graph of f below: sketch a function F such that F ′^ = f.
3.) (10 pts.) Use the table below to estimate
∫ (^6)
0
f (t) dt.
t 0 2 4 6 f (t) 88 45 16 0
4.) (10 pts.) A square-bottomed box with a top has a fixed volume, V. What dimensions minimize the surface area?
7.) (10 pts.) Compute the derivative S′(t) if S(t) = ex^ ln(3x^2 ) − arcsin(cos x).
8.) (10 pts.) Compute all critical points and inflection points of the function g(x) = x +
x
9.) (10 pts.) If f (x) = 3x^2 − 1 and g(x) = x + 1, compute f (g(x)) and g(f (x)).
10.) (10 pts.) Compute the derivative of f (x) =
x algebraically: that is, set up the appropriate limit and use algebra to compute it. You may check your work using the Power Rule, but using the Power Rule alone earns no credit.
