Chapter 14: Q. 33 (page 906)

In Problems $33 - 35$ , find the derivative of each function at the number indicated.
$33. f (x) = - 4 x^{2} + 5$ at $3$

Short Answer

Expert verified

The derivative of $f (x) = - 4 x^{2} + 5$ at $3$ is $f' (3) = - 24$

Step by step solution

Step 1. Find f(3)

To find the derivative we have to use the formula,

$f^{'} (c) = lim_{x \to c} \frac{f (x) - f (c)}{x - c}$

We have been given $f (x) = - 4 x^{2} + 5$ and $c = 3$

Now we have to find $f (c)$ , therefore substitute $c = 3$ in $f (x)$ to find $f (3)$ .

$\begin{aligned} f (x) & = - 4 x^{2} + 5 \\ f (3) & = - 4 (3)^{2} + 5 \\ = - 36 + 5 \\ = - 31 \end{aligned}$

Step 2. Use the derivative formula

Substituting all the values in the derivative formula we get,

$\begin{aligned} f^{'} (3) & = lim_{x \to 3} \frac{f (x) - f (3)}{x - (3)} \\ = lim_{x \to 3} \frac{- 4 x^{2} + 5 - (- 31)}{x - 3} \\ = lim_{x \to 3} \frac{- 4 x^{2} + 36}{x - 3} \\ = lim_{x \to 3} \frac{- 4 (x - 3) (x + 3)}{x - 3} \end{aligned} \begin{aligned} = lim_{x \to 3} (- 4 x - 12) \\ = (- 4 (3) - 12) \\ = - 12 - 12 \\ = - 24 \end{aligned}$

Hence the derivative of $f (x)$ when $c = 3$ is $f^{'} (3) = - 24$ .

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In Problems $33 - 35$ , find the derivative of each function at the number indicated.
$33. f (x) = - 4 x^{2} + 5$ at $3$

Short Answer

Step by step solution

Step 1. Find f(3)

Step 2. Use the derivative formula

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In Problems 33–35, find the derivative of each function at the number indicated.33.f(x)=−4x2+5at3

Short Answer

Step by step solution

Step 1. Find f(3)

Step 2. Use the derivative formula

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Applied Mathematics

Decision Maths

Calculus

Theoretical and Mathematical Physics

Logic and Functions

Study anywhere. Anytime. Across all devices.

In Problems $33 - 35$ , find the derivative of each function at the number indicated.
$33. f (x) = - 4 x^{2} + 5$ at $3$