## Thomas' Calculus 13th Edition

Published by Pearson

# Chapter 3: Derivatives - Section 3.1 - Tangents and the Derivative at a Point - Exercises 3.1 - Page 108: 16

#### Answer

$y=6x-2$

#### Work Step by Step

The question asks us to find the equation of the tangent of the function $h(t)=t^3+3t$ at the point (1,4) Given that we are looking for the equation of a tangent, the first reasonable step would be to find the gradient at the points given, which can be found using the derivative function. Using the power rule: $h(t)=t^3+3t$ $f′(x)=3t^{3-1}+3=3t^2+3$ $f′(1)=3(1)^2+3=6$ Thus, the gradient of that function at the value x=1 is 6. Using this information, we can find the intercept value of the tangent line. The equation of the gradient is given by: $\frac{y_{1}−y_{0}}{x_{1}−x_{0}}=m$ We can now use the points (1,4) and (0,c) (where c is the y-intercept). $\frac{4−c}{1-0}=6$ Solving for c gives us c=-2 Thus, the equation of the tangent at (1,4) is $y=6x-2$

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