Explanation The total change in the dependent variable over the total change in the independent variable is called the average rate of change. The average rate of change between two points of a function is given by: f ( x 2 ) − f ( x 1 ) x 2 − x 1 , Where, ( x 1 , f ( x 1 ) ) is the starting point of the interval and ( x 2 , f ( x 2 ) ) is the endpoint of the interval. The average rate of change of a function can be negative, positive, or even zero just like the slope of the secant line. To know how fast the particle moves at a particular instant of time, we define Instantaneous rate of change. Given: Q ( t ) = t 2 . Formula used: The average rate of change of Q ( t ) with respect to t is given by: Δ Q ( t ) Δ t = Q ( t 2 ) − Q ( t 1 ) t 2 − t 1 Calculation: The average rate of change of Q over the interval [ 1 , t ] is: Δ Q ( t ) Δ t = Q ( t ) − Q ( 1 ) t − 1 = t 2 − ( 1 ) 2 t − 1 = ( t + 1

4th Edition

ISBN: 9781319050740

Author: Jon Rogawski, Colin Adams, Robert Franzosa

Publisher: W. H. Freeman

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Question

Chapter 2.1, Problem 29E

To determine

A formula for the slope of secant line over the interval [1,t] and use it to estimate the slope of tangent at t=1.

Also, repeat formula for the given interval [2,t] and for the slope of the tangent t=2.

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Chapter 2.1, Problem 28E

Chapter 2.1, Problem 30E

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The graph of f' is below. Use it to determine where the local minima and maxima for f are. If there are multiple answers, separate with commas. 2 f'(x) N -5 -4 3-2-1 -1 -2 -3 -4 12 3 4 5 -x Local minima at x Local maxima at x

The graph of f' is below. Use it to determine the intervals where f is increasing. -5-4-32 4- 3 2 1 -2 -3 +x 2 3 4 5

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Chapter 2 Solutions

Calculus: Early Transcendentals

Ch. 2.1 - Prob. 1PQ Ch. 2.1 - Prob. 2PQ Ch. 2.1 - Prob. 3PQ Ch. 2.1 - Prob. 4PQ Ch. 2.1 - Prob. 1E Ch. 2.1 - Prob. 2E Ch. 2.1 - Prob. 3E Ch. 2.1 - Prob. 4E Ch. 2.1 - Prob. 5E Ch. 2.1 - Prob. 6E

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A formula for the slope of secant line over the interval [ 1 , t ] and use it to estimate the slope of tangent at t = 1 . Also, repeat formula for the given interval [ 2 , t ] and for the slope of the tangent t = 2 .

Solution Summary: The author explains the formula for the slope of secant line over the interval left[1,tright] and the tangent's slope. The average rate of change of a function can be negative,

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A formula for the slope of secant line over the interval [1,t] and use it to estimate the slope of tangent at t=1.

Also, repeat formula for the given interval [2,t] and for the slope of the tangent t=2.

Want to see the full answer?

Chapter 2 Solutions

A formula for the slope of secant line over the interval [ 1 , t ] and use it to estimate the slope of tangent at t = 1 . Also, repeat formula for the given interval [ 2 , t ] and for the slope of the tangent t = 2 .

Solution Summary: The author explains the formula for the slope of secant line over the interval left[1,tright] and the tangent's slope. The average rate of change of a function can be negative,

Concept explainers

Videos

A formula for the slope of secant line over the interval [1,t] and use it to estimate the slope of tangent at t=1. Also, repeat formula for the given interval [2,t] and for the slope of the tangent t=2.

Want to see the full answer?

Chapter 2 Solutions

A formula for the slope of secant line over the interval [1,t] and use it to estimate the slope of tangent at t=1.

Also, repeat formula for the given interval [2,t] and for the slope of the tangent t=2.