ws10 - √ x ) 1 / 2 d) f ( x ) = sin ± cos x x ² b) g (...

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WORKSHEET 10 - Fall 1995 1. a) Explain geometrically what it means for a function to be continuous. Do the same for diFerentiability. b) Draw graphs to describe all possible ways a function can fail to be continuous at a point. Also draw graphs showing how a function can fail to be diFerentiable. c) Graph f ( x )= x 1 / 3 . What does your intuitive picture from parts a) and b) say about the continuity and diFerentiability of f ? d) Compute f 0 ( x ) and graph it. Where is f diFerentiable? 2. a) What is the domain of f ( x )= x 2 sin(1 /x )? b) Rede±ne f so that it is continuous everywhere. That is, wherever f is not de±ned give it a value so that it will be continuous. c) Compute f 0 ( x ). What is lim x 0 f 0 ( x )? d) What is f 0 (0)? e) Repeat steps a) through d) for f ( x )= x sin(1 /x ). Explain geometrically why there is a diFerence. 3. Let g ( x )= ax 2 ,x 2 a 2 4 x 2 +3 ,x< 2. What value(s) of a make g continuous? DiFerentiable? 4. DiFerentiate the following: a) f ( x )=(1+ x
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Unformatted text preview: √ x ) 1 / 2 d) f ( x ) = sin ± cos x x ² b) g ( x ) = [( x 2 + 1) 2 + ( x 2 + 1) + 1] 2 e) g ( x ) = (sin 2 x )(sin x 2 )(sin 2 x 2 ) c) h ( x ) = ³ x − 2 x + sin x ´ − 1 f) h ( x ) = sin(cos x ) x 5. a) Prove that the formula for the derivative for an inverse function is ( f − 1 ) ( x ) = 1 f ( f − 1 ( x )) . (Hint: Let g ( x ) = f − 1 ( x ), then f ( g ( x )) = x . DiFerentiate.) b) ²ind f − 1 ( x ) given that f ( x ) = 2 x − 3 x +2 . c) DiFerentiate f − 1 ( x ) from part b) and compare with the derivative you get by applying the formula in part a). 6. ²ill in the table, given that h(x)=f(g(x)). a g ( a ) g ( a ) f ( a ) f ( a ) h ( a ) h ( a ) 1 3 2-2-1/3 1 2-7 2 2 d.n.e-1 4 7. Sketch a graph for f , g , and h satisfying the table in Problem 6....
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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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