Calculus II Quiz

# Calculus II Quiz - 2 x − 1 cosh x(b Find y if y = x...

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MATH 126 QUIZ #1 Solution 01/24/08 Mathias Knape Last Name : Circle discussion section First Name : 11 am 12 pm 1 pm Note: In order to receive full credit, please show all your work (conceptual steps, deFnitions you use, . ..) 1. Find the following limits. (a) lim x 1 1 x +ln x 1+cos πx With L’Hospital it follow that lim x 1 1 x +ln x 1+cos πx 0 0 = lim x 1 1+ 1 x sin( πx ) π 0 0 = lim x 1 1 x 2 π 2 cos πx = 1 π 2 (b) lim x π 2 (tan x ) cos x Let y = (tan x ) cos x ,thenln y =cos x ln(tan x ). At ±rst we compute lim x π 2 ln y = lim x π 2 cos x ln(tan x ) = lim x π 2 ln(tan x ) 1 cos x = lim x π 2 1 tan x sec 2 x sec x tan x = lim x π 2 1 tan 2 x sec x = lim x π 2 cos x sin 2 x =0 . Returning to the original problem we have that lim x π 2 (tan x ) cos x = lim x π 2 e ln y =1 . 2. (a) Di²erentiate f ( x )=cosh 1 (sinh x ) . f 0 ( x )= 1 p sinh

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Unformatted text preview: 2 x − 1 cosh x (b) Find y if y = x + arctan y . Using implicit di±erentiation we obtain dy dx = 1 + 1 1 + y 2 dy dx . Solving for dy dx we get dy dx = 1 + 1 y 2 . (c) Prove that d dx (sech − 1 x ) = − 1 x √ 1 − x 2 . (The identity 1 − tanh 2 y = sech 2 y might be helpful.) Let y = sech − 1 x . Then sech y = x . Implicit di±erentiation leads to − sech y tanh y dy dx = 1 . Then dy dx = − 1 sech y tanh y = − 1 sech y p 1 − sech 2 y = − 1 x √ 1 − x 2 . In the second to last equation we used that tanh y > 0 because y > 0 ....
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Calculus II Quiz - 2 x − 1 cosh x(b Find y if y = x...

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