ws11 - ( x ) + v ( x )) d) f ( x ) = g ( u ( x ) /v ( x ))...

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WORKSHEET 11 - Fall 1995 1. The curve y = ax 2 + bx + c passes through the point (1 , 2) and is tangent to the line y = x at the origin. Find a , b ,and c . 2. If f is even (or odd), is f 0 necessarily one or the other? Construct a proof. 3. A particle travels along a line whose distance is one meter. At any time t seconds after a certain moment ( t = 0), the particle is at the point s ( t ) on the line, with s ( t )=4 t t 2 . a) What is the location of the particle at time t =3? b) What is its speed at time t =4? c) At what time(s) is the particle at rest? d) For which values of t is the particle moving to the right? e) For which values of t is the particle moving to the left and slowing down? 4. Let F ( x )= g ( x ) h ( x ). a) Compute F 0 , F 0 , F 0 ,and F 0 ( x )intermso f g , h , and their derivatives. b) Compute ( A + B ) n for n =1 , 2 , 3, and 4. c) Compare. 5. Let f ( x )= x 3 +3 x 2 +3 x +5. Does f have an inverse? How do you know? If f has an inverse, determine f 1 (5), ( f 1 ) 0 (5), f 1 (12), and ( f 1 ) 0 (12). 6. Suppose that g , u ,and v are di±erentiable functions. Find f 0 in terms of g , u , v , g 0 , u 0 ,and v 0 in each of the following cases. a) f ( x )= g ( u ( v ( x ))) c) f ( x )= g ( u ( x ) v ( x )) b) f ( x )= g (
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Unformatted text preview: ( x ) + v ( x )) d) f ( x ) = g ( u ( x ) /v ( x )) Comments: Should I have written f ( x ) = g ( u ( v ( x ))) in a) above as f ( x ) = g u v ( x ) ! to make the question clearer and more fun typographically? By the way, real computations similar to these denitely arise in elds as widely separated as thermodynamics and economics. 7. Let f be the function such that f ( x ) is the value of the sine of an angle which measures x degrees . (NOTE: For most values of x , f ( x ) 6 = sin x . Why not?) Let g be dened similarly for cosine. a) Express f and g in terms of sin and cos. b) What is df dx ? What is dg dx ? (Hint: Use part a) and the chain rule.) c) Express df dx and dg dx in terms of f ( x ) and g ( x ). (No mention of sin or cos allowed.) d) Is it still true that ( f ( x )) 2 + ( g ( x )) 2 = 1? e) Why dont we use degrees in calculus?...
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This note was uploaded on 04/11/2011 for the course MATH 1400 taught by Professor Grether during the Spring '08 term at North Texas.

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