## Calculus (3rd Edition)

Published by W. H. Freeman

# Chapter 7 - Exponential Functions - 7.3 Logarithms and Their Derivatives - Exercises - Page 343: 55

#### Answer

$$y= (3^{18} 9\ln 3) \ t+3^{18}(1- 18\ln 3).$$

#### Work Step by Step

Let $f(t)=3^{9t}$, then $f'(t)=3^{9t}9 \ln 3$ and the slope of $f$ at $t=2$ is given by $$m=f'(2)=3^{18} 9\ln 3.$$ The equation of the tangent line is $$y= (3^{18} 9\ln 3) \ t+c.$$ Since the function and the tangent line coincide at $t=2$, we have $$c=3^{18}- (3^{18} 9\ln 3) \ (2)=3^{18}(1- 18\ln 3) .$$ Finally, we get $$y= (3^{18} 9\ln 3) \ t+3^{18}(1- 18\ln 3).$$

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