Calculus: Graphical, Numerical, Algebraic

3rd Edition

ISBN: 9780133688399

Author: Finney, Ross L.

Publisher: Pearson/Prentice Hall

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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 5.2, Problem 57E

(a)

To determine

To find: The find the region where the f is increasing, decreasing or has the local extrema.

(a)

Expert Solution

Answer to Problem 57E

The region where the f is increasing, decreasing or has the local extrema is obtained.

Explanation of Solution

Given:

The given table is shown in Table 1

Table 1

x	f′(x)	x	f′(x)
-2	7	0.25	4.81
-1.75	4.19	0.5	-4.25
-1.5	1.75	0.75	-3.31
-1.25	-0.31	1	-2
-1	-2	1.25	-0.31
-0.75	-3.31	-1.5	1.75
-0.5	-4.25	1.75	4.19
-0.25	-4.81	2	7
0	-5

Calculation:

The graph for the values given in the table is shown in Figure 1

Calculus: Graphical, Numerical, Algebraic, Chapter 5.2, Problem 57E

Figure 1

Consider that the f is continuous on [−2,2] and differentiable at (−2,2) .

Then, if the function has the maximum value or the local minimum value at the interior point c of the domain and if f′ exists at c then f′(c)=0 .

Consider that f is continuous on [a,b] and it is differentiable at (a,b) .

Consider that f′>0 is continuous on [a,b] and it is differentiable at [a,b] .

Consider that f′<0 is continuous on [a,b] and it is differentiable at [a,b] .

Consider that for the given table x=−2 to −1.5 then the value of f′(x) is positive and the curve meets the x axis at the points x=−1.3 .

From this it is clear that the function increases on the interval [1.3,1.3] .

Consider that for the given table x=−1.25 to 1.25 then the value of f′(x) is negative and the curve meets the x axis at the points x=−1.3 and x=1.3 .

From this it is clear that the function decrease on the interval [−1.3,1.3] .

Thus, the local maxima for the given function is at x=1.3 and the minima is at x=−1.3 .

(b)

To determine

To find: The quadratic regression of the equation for the data in the table and superimpose the graph on the scatter plot of the data.

(b)

Expert Solution

Answer to Problem 57E

The required regression equation is y=3x2−5 and the regression line on the graph is shown in Figure 1

Explanation of Solution

Calculation:

From the table 1 when the values are entered in the calculator the regression equation is obtained as,

y=3x2−5

(c)

To determine

To find: The formula for the f that satisfies the f(0)=0 .

(c)

Expert Solution

Answer to Problem 57E

The function is f(x)=x3−5x .

Explanation of Solution

Calculation:

Consider the regression equation is,

y=3x2−5

Then,

f(x)=x3−5x+Cf(0)=(0)3−5(0)+CC=0

Then the function is f(x)=x3−5x .

Chapter 5.2, Problem 56E

Chapter 5.2, Problem 58E

Chapter 5 Solutions

Calculus: Graphical, Numerical, Algebraic

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