The Derivative 61. Definition If f is a function, the statement f has derivative D at the pointc in the domain of f means that (a) c is a limit point of the domain of f, and is also in the domain of f. (b) if (a is any sequence converging to c such that an c is in the domain of f for all n E N, then f(an)-f(c) is a sequence that converges to D. Notation: WWe denote the derivative of f at the point c by the familiar f'(c). 62. Use the definition to prove that f'(--3) -6, where f(z) -= 2+2 63 Une the definition to prove that f'() 2x, for all z eR, if f(x) 64. Suppose f has a derivative at the pointc Prove f is also continuous at c Find an example of a function that is contimous at some point c but not differentiable at c. Then prove your assertion. Assume that f has domain containing (a,, and f has a derivative at c e (a,b). Further assume that f(r) S f(c) for all z e (a, b). Prove f (c) - 0. (Hint: Try to show that f'(c) S0 and f(c) 2 0.) 67. Assume that f.g are differentiable functions on the interval (a,b). Define h(r) f(r) + g(). Prove that A'(z) f()+ g(a). 68. Let a be a constant, and let f be a differentiable function on the set (a, b). Let af(a) for all r e la, b. Prove M(2) -= af(z), for all e (a, bl A(r) 69. Prove the product rule for derivatives. State the hypotheses and then prove the theo- rem. 70. Prove the chain rule for derivatives. State the hypotheses and then prove the theorem.
The Derivative 61. Definition If f is a function, the statement f has derivative D at the pointc in the domain of f means that (a) c is a limit point of the domain of f, and is also in the domain of f. (b) if (a is any sequence converging to c such that an c is in the domain of f for all n E N, then f(an)-f(c) is a sequence that converges to D. Notation: WWe denote the derivative of f at the point c by the familiar f'(c). 62. Use the definition to prove that f'(--3) -6, where f(z) -= 2+2 63 Une the definition to prove that f'() 2x, for all z eR, if f(x) 64. Suppose f has a derivative at the pointc Prove f is also continuous at c Find an example of a function that is contimous at some point c but not differentiable at c. Then prove your assertion. Assume that f has domain containing (a,, and f has a derivative at c e (a,b). Further assume that f(r) S f(c) for all z e (a, b). Prove f (c) - 0. (Hint: Try to show that f'(c) S0 and f(c) 2 0.) 67. Assume that f.g are differentiable functions on the interval (a,b). Define h(r) f(r) + g(). Prove that A'(z) f()+ g(a). 68. Let a be a constant, and let f be a differentiable function on the set (a, b). Let af(a) for all r e la, b. Prove M(2) -= af(z), for all e (a, bl A(r) 69. Prove the product rule for derivatives. State the hypotheses and then prove the theo- rem. 70. Prove the chain rule for derivatives. State the hypotheses and then prove the theorem.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.3: Geometric Sequences
Problem 81E
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Use definition in #61 to solve the question for #66
Transcribed Image Text:The Derivative
61. Definition If f is a function, the statement f has derivative D at the pointc in the
domain of f means that
(a) c is a limit point of the domain of f, and is also in the domain of f.
(b) if (a is any sequence converging to c such that an c is in the domain of f
for all n E N, then
f(an)-f(c)
is a sequence that converges to D.
Notation: WWe denote the derivative of f at the point c by the familiar f'(c).
62. Use the definition to prove that f'(--3) -6, where f(z) -= 2+2
63 Une the definition to prove that f'()
2x, for all z eR, if f(x)
64. Suppose f has a derivative at the pointc Prove f is also continuous at c
Find an example of a function that is contimous at some point c but not differentiable
at c. Then prove your assertion.
Assume that f has domain containing (a,, and f has a derivative at c e (a,b).
Further assume that f(r) S f(c) for all z e (a, b). Prove f (c) - 0. (Hint: Try to
show that f'(c) S0 and f(c) 2 0.)
67. Assume that f.g are differentiable functions on the interval (a,b). Define h(r)
f(r) + g(). Prove that A'(z) f()+ g(a).
68. Let a be a constant, and let f be a differentiable function on the set (a, b). Let
af(a) for all r e la, b. Prove M(2) -= af(z), for all e (a, bl
A(r)
69. Prove the product rule for derivatives. State the hypotheses and then prove the theo-
rem.
70. Prove the chain rule for derivatives. State the hypotheses and then prove the theorem.
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