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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?
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)?
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 ...
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- Fall '07
- Math, Sin, pts, Inverse function, Logarithm, BRIAN OSSERMAN