Exercise 5.2.6. Let g be defined on an interval A, and let cЄ A. (a) Explain why g'(c) in Definition 5.2.1 could have been given by g'(c) = lim g(c+ h) - g(c) h→0 h (b) Assume A is open. If g is differentiable at c E A, show g'(c) = lim g(c+h)-g(c-h) h→0 2h Definition 5.2.1 (Differentiability). Let g: AR be a function defined on an interval A. Given cЄ A, the derivative of g at c is defined by g'(c) = lim g(x)-g(c) x c X-C provided this limit exists. In this case we say g is differentiable at c. If g' exists for all points cЄ A, we say that g is differentiable on A.
Exercise 5.2.6. Let g be defined on an interval A, and let cЄ A. (a) Explain why g'(c) in Definition 5.2.1 could have been given by g'(c) = lim g(c+ h) - g(c) h→0 h (b) Assume A is open. If g is differentiable at c E A, show g'(c) = lim g(c+h)-g(c-h) h→0 2h Definition 5.2.1 (Differentiability). Let g: AR be a function defined on an interval A. Given cЄ A, the derivative of g at c is defined by g'(c) = lim g(x)-g(c) x c X-C provided this limit exists. In this case we say g is differentiable at c. If g' exists for all points cЄ A, we say that g is differentiable on A.
Chapter3: Functions
Section3.3: Rates Of Change And Behavior Of Graphs
Problem 2SE: If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local...
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Transcribed Image Text:Exercise 5.2.6. Let g be defined on an interval A, and let cЄ A.
(a) Explain why g'(c) in Definition 5.2.1 could have been given by
g'(c) = lim
g(c+ h) - g(c)
h→0
h
(b) Assume A is open. If g is differentiable at c E A, show
g'(c) = lim
g(c+h)-g(c-h)
h→0
2h
Transcribed Image Text:Definition 5.2.1 (Differentiability). Let g: AR be a function defined
on an interval A. Given cЄ A, the derivative of g at c is defined by
g'(c) = lim
g(x)-g(c)
x c
X-C
provided this limit exists. In this case we say g is differentiable at c. If g' exists
for all points cЄ A, we say that g is differentiable on A.
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