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We say f : R –→ R is even if f(-x) = f(x) for all x. We say f is odd if f(-x) = -f(x) for all x. (a) Prove that if f is even and differentiable then f' is odd. Demonstrate this result graphically using a generic example function. (b) Prove that if f is odd and differentiable then f' is even. Demonstrate this result graphically using a generic example function.

We say f : R –→ R is even if f(-x) = f(x) for all x. We say f is odd if f(-x) = -f(x) for all x. (a) Prove that if f is even and differentiable then f' is odd. Demonstrate this result graphically using a generic example function. (b) Prove that if f is odd and differentiable then f' is even. Demonstrate this result graphically using a generic example function.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section5.3: The Natural Exponential Function
Problem 52E
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We say f : R –→R is even if f(-x) = f(x) for all x. We say f is odd if f(-x) = -f(x) for all x. (a) Prove that if f is even and differentiable then f' is odd. Demonstrate this result graphically using a generic example function. (b) Prove that if f is odd and differentiable then f' is even. Demonstrate this result graphically using a generic example function.
Transcribed Image Text:We say f : R –→R is even if f(-x) = f(x) for all x. We say f is odd if f(-x) = -f(x) for all x. (a) Prove that if f is even and differentiable then f' is odd. Demonstrate this result graphically using a generic example function. (b) Prove that if f is odd and differentiable then f' is even. Demonstrate this result graphically using a generic example function.
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