Hmw3 - √ 3 7 Let C be a curve given by a parametric equation x t,y t where x(1 =-2 y(1 = 4 x 00(1 = 5 y 00(1 =-3 Find d 2 y dx 2 at t = 1 8 Let f

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Due on the 1st lecture of next week, Nov 17 - Nov 21. Homework 3 1. Determine whether the following statements are true or false. Prove if true and give an example if false. (a) If f ( x ) takes on every value between f ( a ) and f ( b ) on the interval [ a,b ] then f ( x ) is continuous. (b) If f ( x ) is continuous and lim x 0 f ( x ) x = - 10 then f ( x ) is differentiable. (c) If f 0 has a zero in ( a,b ) then f must have either a maximum or a minimum in ( a,b ). 2. Show that the equation f ( x ) = f 0 ( x ) has three solutions for f ( x ) = x 3 - 3 x . 3. Find the points where the tangent line to the parabola y = x 2 passes through the point (2 , - 5). 4. Show that the function f ( x ) = x | x | , x 6 = 0 1 , x = 0 is not the derivative of any function. 5. Show that f ( x ) = x, x 0 x + 1 , x < 0 is not the derivative of any function. 6. Find the points on the unit circle where the tangent line has slope 1
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Unformatted text preview: / √ 3. 7. Let C be a curve given by a parametric equation ( x ( t ) ,y ( t )) where x (1) =-2, y (1) = 4, x 00 (1) = 5, y 00 (1) =-3. Find d 2 y dx 2 at t = 1. 8. Let f ( x ) = x 2 sin ± 1 x ¶ when x 6 = 0 and f (0) = 0. Show that f ( x ) exists for all x ∈ R but f ( x ) is not continuous. 9. Assume that f ( x ) is differentiable and that f ( x ) 6 = 0 for any x . (a) Find lim x → f ( (-1 + 2 x ) 3 + 2 )-f (1) f (1)-f (tan( π/ 4-x )) . (b) Given f (2) = 7, find lim x → f (2 + 3 x )-f (2-5 x ) 11 x . 10. Assume that f ( x ) is differentiable on [-4 , 4], f (4) = 0 and f ( x ) is even. Show that f ( a ) = f ( a + 2) for some a ∈ [0 , 2]....
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This note was uploaded on 10/29/2010 for the course MATH 111 taught by Professor Ahmetguloglu during the Spring '10 term at Bilkent University.

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