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Math 251 Quiz: Derivatives and Inverse Functions, Quizzes of Calculus

Material Type: Quiz; Class: Calculus I >4; Subject: Mathematics; University: University of Oregon; Term: Fall 2006;

Typology: Quizzes

Pre 2010

Uploaded on 07/23/2009

koofers-user-9sk
koofers-user-9sk 🇺🇸

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Quiz for Math 251 (CRN 15056), Week 7
Note: Show your work! These are the important basis for grading.
Name
1. True or false. (25 points)
a) If g(x) = f1(x), then g0(x) = 1
f0(x). T F
b) (tan x)0=1
1 + tan2x. T F
c) (tan1x)0=1
1 + tan2x. T F
d) d4
dx4sin x= sin x. T F
e) d2
dx2(f·g) = ( d2
dx2f)·g+f·(d2
dx2g), here fand gare abbreviations for
functions f(x) and g(x).
T F
2. Find the derivative of y=(x+ 1)2·(x2)
3x1·x3. (25 points)
1
pf2

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Quiz for Math 251 (CRN 15056), Week 7

Note: Show your work! These are the important basis for grading.

Name

  1. True or false. (25 points)

a) If g(x) = f −^1 (x), then g′(x) =

f ′(x)

. T F

b) (tan x)′^ =

1 + tan^2 x

. T F

c) (tan−^1 x)′^ =

1 + tan^2 x

. T F

d)

d^4 dx^4 sin x = sin x. T F

e) d^2 dx^2

(f · g) = ( d^2 dx^2

f ) · g + f · ( d^2 dx^2

g), here f and g are abbreviations for functions f (x) and g(x).

T F

  1. Find the derivative of y =

(x + 1)^2 · (x − 2) 3

x − 1 · x^3

. (25 points)

  1. f (x) = sin sin x for − 1 ≤ x ≤ 1. On the domain (− 1 , 1), f (x) is a one-to-one function, thus we may define f −^1 (x), just let g(x) = f −^1 (x). As sin(sin 0) = 0, it is obvious that g(0) = f −^1 (0) = 0. The question: Find the value of g′(0). (i.e, find the derivative of inverse function of f at 0) (25 points).
  2. Use implicit differentiation to find y′^ at (0, 0) for

x · sin y + (x + 1)^2 · ey^ = 1

(25 points)