Calculus, 10th Edition (Anton)

Published by Wiley
ISBN 10: 0-47064-772-8
ISBN 13: 978-0-47064-772-1

Chapter 6 - Exponential, Logarithmic, And Inverse Trigonometric Functions - 6.3 Derivatives Of Inverse Functions; Derivatives And Integrals Involving Exponential Functions - Exercises Set 6.3: 46

Answer

True.

Work Step by Step

The derivative, or slope of the tangent line, of $f(x)=b^{x}$ is $f'(x)=b^{x}ln(b)$. If the slope of tangent line at $x=0$ is $1$, we know that $$f'(0)=1$$ Replacing $f'(0)$ with the derivative at $x=0$ yields $$b^{0}ln(b)=1$$ $$ln(b)=1$$ Solving for $b$ by raising each side of the equation to the $e$ yields $b=e$. An exponential function with base $e$ is the natural exponential function.
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