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Calculus I EXAM 2
MATH 105 March 16, 2011
Name:
Your grade is based on correctness, completeness, and clarity on each exercise. Explain all answers completely. You may use a calculator, but no notes, books, or other students. Good luck!
1.) (10 pts.) Use the given table to compute and evaluate the derivatives of the functions below.
x f (x) f ′(x) g(x) g′(x) 1 3 7 9 5 9 5 3 1 7
a.) Compute the derivative of y = xf (x), then evaluate when x = 1.
b.) Compute the derivative of y = f (g(x)), then evaluate when x = 1.
2.) (15 pts.) Compute the derivative of y =
x^2 + 4x + 3 x + 3 two ways.^ Then compare your answers.
a.) (5 pts.) Do not simplify first. Use the Quotient Rule. Simplify AFTER computing the derivative.
b.) (5 pts.) Simplify the fraction FIRST. Then use the Power Rule to compute the deriva- tive.
c.) (5 pts.) Compare your two fully simplified final answers. Are they the same? Should they be?
4.) (15 pts.)
a.) (5 pts.) Use a reference triangle to write cos(arctan(5x)) as an algebraic expression, that is, one not involving any trigonometric functions. (Note: you are not being asked to compute a derivative here.)
b.) (5 pts.) Compute the derivative of ln(arcsin(x^2 )).
c.) (5 pts.) Compute an antiderivative of g(x) = e
x (^) + sec (^2) x + 2x ex^ + tan x + x^2 + π.
5.) (15 pts.)
a.) (5 pts.) Sketch a graph (it is fine to use your calculator) of y = 4010 xx^44 +4+8xx^22 −+1^1 and use your graph to make a guess about lim x→∞^40 x
(^4) + 4x (^2) − 1 10 x^4 + 8x^2 + 1. (Be sure to show^ how^ you are using your graph to make your guess.)
b.) (5 pts.) Use an algebraic or calculus-based technique to confirm your guess from part (a).
c.) (5 pts.) Use L’Hˆopital’s Rule to compute lim x→ 0 e
x (^) − 1 x^2 + 3x. Be sure to confirm that this is a case in which you can use L’Hˆopital’s Rule.
7.) (15 pts.) Consider the function y = 12 x + sin x on the interval [0, 2 π].
a.) (5 pts.) At which x-values, from 0 to 2π, does y have stationary points? Use calculus to produce your answer, and give the x-values exactly, not as decimal approximations.
b.) (5 pts.) Demonstrate, using the First Derivative Test and/or the Second Derivative Test, whether each of your stationary points from part (a) is a local maximum or a local minimum.
c.) (5 pts.) How many points of inflection does y have for x-values in the interval [0, 2 π]? Use calculus to support your answer, and as in part (a), give all x-values exactly.
BONUS: The mathematical definition of continuity at a point “a” involves the concept of a limit. Explain this connection. (You can state the definition of continuity at a point, or you can give a more intuitively-worded explanation that involves limits.) You may use the back of this page if you would like more space to write.
