Practice Exam2 solutions

Practice Exam2 solutions - Math 21A 1(40 pts Brian Osserman...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 21A 1 (40 pts.) Brian Osserman Practice Exam 1 Dierentiate the following functions. Show your work if you wish to receive partial credit. You do not need to simplify your answers. (a) f (x) = 3x+7 x4 +1 . f (x) = -3x4 - 14x3 + 3 . (x4 + 1)3/2 Answer: (b) f (x) = x2 +1 tan-1 (x2 +1) . Answer: f (x) = 2x((x4 + 2x2 + 2) tan-1 (x2 + 1) - (x2 + 1)) . (tan-1 (x2 + 1))2 (x4 + 2x2 + 2) (c) f (x) = e2x-1 x + 1. Answer: f (x) = e2x-1 (4x + 5) . 2 x+1 (d) f (x) = sec2 (x + sin x). Answer: f (x) = 2(1 + cos x) sec2 (x + sin x) tan(x + sin x). (e) f (x) = (ln x)10 . Answer: f (x) = 10(ln x)9 . x 2 (10 pts.) Suppose u(x) is a dierentiable function of x, and let f (x) = xu(x) , for x > 0. Find f (x), in terms of x, u(x), and u (x). Hint: take the natural log of both sides and use implicit dierentiation. Answer: f (x) = xu(x) (u (x) ln x + 3 (20 pts.) u(x) ). x Consider the parametric curve (- t + 1, 3t). (a) Find the tangent line to the curve at the point corresponding to t = 3. Answer: y = -2x - 1. (b) Find d2 y/(dx)2 at the same point. Answer: d2 y/(dx)2 = -1/3 4 (10 pts.) at t = 3 Find the equation of the line tangent to the curve dened implicitly by 2 sin-1 y = x2 at the point ( Answer: y = x 2 , 2 1 ) 2 . + 2- . 2 2 5 (20 pts.) Consider the function f (x) = x sin 1 : x = 0 x 0 : x=0 . (a) Is f (x) continuous at x = 0? Why or why not? Answer: 1 Yes. x sin x is sandwiched by x and -x. (b) Is f (x) dierentiable at x = 0? Why or why not? If so, what is f (0)? Answer: 1 No. x sin x hits both y = x and y = -x inside (-, ) for any > 0, so the slope of the secants hits both 1 and -1 for any . 2 ...
View Full Document

This homework help was uploaded on 04/07/2008 for the course MATH 21A taught by Professor Osserman during the Fall '07 term at UC Davis.

Page1 / 2

Practice Exam2 solutions - Math 21A 1(40 pts Brian Osserman...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online