Chapter 14: Q, 14 (page 907)

For the function $f (x) = 4 x^{2} - 11 x - 3$
(a) Find the derivative of f at $x = 2$ .
(b) Find the equation of the tangent line to the graph of $f$ at the point $(2, - 9)$ .
(c) Graph $f$ and the tangent line.

Short Answer

Expert verified

The derivative of the $f (2) = 5$ and the equation at $(2, - 9) = y = 5 x - 19$

Step by step solution

Part(a) Step 1, Given Information

We are given a function $f (x) = 4 x^{2} - 11 x - 3$

We need to find the derivative of the function at $x = 2$ $x = 2 .$

Part(a) Step 2. Explanation

We know,

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

Also,

$f (2) = 4 (2)^{2} - 11 (2) - 3 = - 9$

Then, using the formula,

$f^{'} (2) = \lim_{x \to 2} \frac{4 x^{2} - 11 x - 3 - (- 9)}{x - 2} = \lim_{x \to 2} \frac{4 x^{2} - 11 x + 6}{x - 2} = \lim_{x \to 2} \frac{(4 x - 3) (x - 2)}{x - 2} = \lim_{x \to 2} (4 x - 3) = 4 (2) - 3 = 5$

Part(b) Step 1. Explanation

We know,

$m_{tan} = \lim_{x \to c} \frac{f (x) - f (c)}{x - c} w h e r e m = s l o p e .$

On substituting the point $(2, - 9)$

$m_{tan} = \lim_{x \to 2} \frac{4 x^{2} - 11 x - 3 - (- 9)}{x - 2} = \lim_{x \to 2} \frac{4 x^{2} - 11 x + 6}{x - 2} = \lim_{x \to 2} \frac{(4 x - 3) (x - 2)}{x - 2} = \lim_{x \to 2} 4 x - \lim_{x \to 2} 3 = 5$

The equation of a tangent line is,

$y - f (c) = m_{\tan} (x - c)$

On substituting,

$y - (- 9) = 5 (x - 2) y + 9 = 5 x - 10 y = 5 x - 19$

Part(c) Step 1. Explanation

The equation of the tangent line for the function

$f (x) = 4 x^{2} - 11 x - 3$ at $(2, - 9)$ is $y = 5 x - 19$ .

Plotting both the lines, we get the graph,

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For the function $f (x) = 4 x^{2} - 11 x - 3$
(a) Find the derivative of f at $x = 2$ .
(b) Find the equation of the tangent line to the graph of $f$ at the point $(2, - 9)$ .
(c) Graph $f$ and the tangent line.

Short Answer

Step by step solution

Part(a) Step 1, Given Information

Part(a) Step 2. Explanation

Part(b) Step 1. Explanation

Part(c) Step 1. Explanation

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For the function f(x)=4x2-11x-3(a) Find the derivative of f at x=2. (b) Find the equation of the tangent line to the graph of fat the point(2,-9). (c) Graph fand the tangent line.

Short Answer

Step by step solution

Part(a) Step 1, Given Information

Part(a) Step 2. Explanation

Part(b) Step 1. Explanation

Part(c) Step 1. Explanation

One App. One Place for Learning.

Most popular questions from this chapter

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Study anywhere. Anytime. Across all devices.

For the function $f (x) = 4 x^{2} - 11 x - 3$
(a) Find the derivative of f at $x = 2$ .
(b) Find the equation of the tangent line to the graph of $f$ at the point $(2, - 9)$ .
(c) Graph $f$ and the tangent line.