Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Main points of this exam paper are: Implicit Differentiation, Equation, Tangent Line, Graph, Quantities, Function, Quotient Rule, Simplify, Power Rule, Simplify
Typology: Exams
1 / 6
This page cannot be seen from the preview
Don't miss anything!
MATH 105 TEST 2 October 28, 2002
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.) (10 pts.) Use implicit differentiation to find the equation of the tangent line to the graph of
(x + y)^2 = (2x + 1)^3
at the point (0, 1).
2.) (10 pts.) a.) Find h′(t) if h(t) = tπ 3
b.) Find g′(t) if g(t) =
( 1 e
)t
3.) (10 pts.) a.) Find
d dx (cos(sin(2x)))
b.) Find d dθ
( ecos^ θ^ + esin^ θ
)
5.) (10 pts.) a.) Find d dx
( ln(x^2 ex)
) .
b.) Find d dx
(3 arctan x + arcsin(πx))
6.) (10 pts.) What is the local linearization of ex^ near x = 0? Sketch both the curve f (x) = ex and its tangent line to answer the question: are the approximated values greater than or less than the actual values of f (x), for x near 0?
7.) (10 pts.) Let f (x) and g(x) be two functions. Values of f (x), f ′(x), g(x), and g′(x) for x = 0, 1, and 2 are given in the table below. Use the information in the table to answer the questions that follow.
x f (x) f ′(x) g(x) g′(x) 0 1 − 1 2 5 1 − 1 2 4 0 2 7 3 11 0. 5
a.) If H(x) = ef^ (x)^ + πx, then what is H′(0)?
b.) If K(x) = f (g(x)), then what is K′(0)?
8.) (10 pts.) Use L’Hopital’s Rule to find lim x→ 0
sin x cos x x
. Remember: you still have to justify using
L’Hopital’s Rule BEFORE you can use it.
