Explanation Given: f ( x ) = 4 1 + 2 x Formula Used: The derivative of the function f ( x ) is given by the limit f ' ( x ) = lim h → 0 f ( x + h ) − f ( x ) h … ( 1 ) Equation of the tangent line is: y = f ( a ) + f ' ( a ) ( x − a ) … ( 2 ) Calculation: (a) The graph of f ( x ) = 4 1 + 2 x over [ − 2 , 2 ] is as shown below: Now, zooming in the graph near x = 0 , we get: The slope of the graph near x = 0 can be calculated using the equation (1) as below: f ' ( 0 ) = lim h → 0.5 f ( 0 + h ) − f ( 0 ) h f ' ( 0 ) = f ( 0 + 0.5 ) − f ( 0 ) 0.5 ≈ 1.65 − 2 0...

3rd Edition

ISBN: 9781319030728

Author: Rogawski

Publisher: MAC HIGHER

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Concept explainers

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

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Question

Chapter 3.1, Problem 49E

To determine

(a) We need to plot f over $[- 2, 2]$ and estimate the slope $f' (0)$ by zooming in the graph near $x = 0$ .

(b) We need to find an approximate equation to the tangent line at $x = 0$ . Also, we need to plot the tangent line and $f (x) = \frac{4}{1 + 2^{x}}$ on the same axes.

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Chapter 3.1, Problem 48E

Chapter 3.1, Problem 50E

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Chapter 3 Solutions

Calculus: Early Transcendentals

Ch. 3.1 - Prob. 1PQ Ch. 3.1 - Prob. 2PQ Ch. 3.1 - Prob. 3PQ Ch. 3.1 - Prob. 4PQ Ch. 3.1 - Prob. 5PQ Ch. 3.1 - Prob. 1E Ch. 3.1 - Prob. 2E Ch. 3.1 - Prob. 3E Ch. 3.1 - Prob. 4E Ch. 3.1 - Prob. 5E

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(a) We need to plot f over [ − 2 , 2 ] and estimate the slope f ' ( 0 ) by zooming in the graph near x = 0 . (b) We need to find an approximate equation to the tangent line at x = 0 . Also, we need to plot the tangent line and f ( x ) = 4 1 + 2 x on the same axes.

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(a) We need to plot f over [−2,2] and estimate the slope f'(0) by zooming in the graph near x=0. (b) We need to find an approximate equation to the tangent line at x=0. Also, we need to plot the tangent line and f(x)=41+2x on the same axes.

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(a) We need to plot f over $[- 2, 2]$ and estimate the slope $f' (0)$ by zooming in the graph near $x = 0$ .

(b) We need to find an approximate equation to the tangent line at $x = 0$ . Also, we need to plot the tangent line and $f (x) = \frac{4}{1 + 2^{x}}$ on the same axes.