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Calculus test questions from math 105, including finding derivatives using the quotient rule and product rule, computing limits, finding tangent lines, and solving related problems. Students are required to show all their steps for partial credit.
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MATH 105 TEST 2 March 5, 2003
Name:
Remember that final answers are not as important as how you get there. Show all your steps clearly so you will be eligible for the most partial credit. Simplify arithmetic quantities completely. Good luck!
1.) (20 pts.) For f (x) = 3 − x^2 3 + x^2
, find f ′(x) in two ways:
a.) use the Quotient Rule;
b.) rewrite as f (x) = (3 − x^2 )(3 + x^2 )−^1 and use the Product Rule.
2.) (10 pts.) Compute the limit (^) xlim→∞ x ex^
. Show all necessary steps.
3.) (10 pts.) What is the tangent line approximation to ex^ near x = 0?
6.) (10 pts.) If f (2) = 5, f ′(2) = 6, g(2) = 7, and g′(2) = 8, find a.) h′(2) if h(x) = f (x) · g(x);
b.) k′(2) if k(x) =
f (x) g(x)
7.) (10 pts.) Compute the 300th derivative of y = sin x.
8.) (10 pts.) Compute h′(z) if h(z) =
( b a + z^2
) 4
. Note: z is the variable; a and b are constants.
9.) (10 pts.) Compute f ′(t) if f (t) = ln(ln t) + ln(ln 2).
BONUS: Use the chain rule to compute
d dx (arcsin x). (You may work on the back of the page.)
