Chapter 3.3, Problem 25E

To determine

The number which is the slope of the line tangent to the curve at $x=3$ .

Answer to Problem 25E

The correct answer is iii.

Explanation of Solution

Given information:

The curve: $y=x^2+5x$

The numbers:

i. 24

ii. $-\frac{5}{2}$

iii. 11

iv. 8

Formula used:

Power rule:

  $\frac{d}{dx}{x}^n=n{x}^{n-1}$

Calculation:

The curve is $y=x^2+5x$ .

To find the slope of the tangent line to the given curve, differentiate the above curve.

Use the power rule $\frac{d}{dx}{x}^n=n{x}^{n-1}$ .

  $\begin{array}{c}\frac{dy}{dx}=\frac{d}{dx}\left(x^2+5x\right)\\ =\frac{d}{dx}{x}^2+\frac{d}{dx}5x\\ =2x+5\end{array}$

So, $y^{\prime }=2x+5$

Substitute 3 for x in the above derivative.

  $\begin{array}{c}{y}^{\prime }=2\left(3\right)+5\\ =6+5\\ =11\end{array}$

Thus, the slope of the tangent line is 11.

Hence, the correct answer is iii.