BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 2.7, Problem 45E
To determine

To find: The first and second derivative of the function f(x)=3x2+2x+1.

Expert Solution

Answer to Problem 45E

The first and second derivatives of the function are, f(x)=6x+2 and f(x)=6.

Explanation of Solution

Formula used:

The derivative of a function f , denoted by f(x), is

f(x)=limh0f(x+h)f(x)h (1)

Calculation:

Obtain the derivative of the function f(x).

Compute f(x) by using the equation (1),

f(x)=limh0f(x+h)f(x)h=limh03(x+h)2+2(x+h)+1{3x2+2x+1}h=limh0(3x2+3h2+6xh)+(2x+2h)+1{3x2+2x+1}h=limh03x2+3h2+6xh+2x+2h+13x22x1h

Simplify the numerator and to obtain the value of f(x),

f(x)=limh03h2+6xh+2hh=limh0h(3h+6x+2)h

Since the limit h approaches zero but not equal to zero, cancel the common term h from both the numerator and the denominator,

f(x)=limh0(3h+6x+2)=3(0)+6x+2=6x+2

Thus, the first derivative of the function is, f(x)=6x+2.

Obtain the second derivative of the function.

Compute f(x) by using the equation (1),

f(x) =(f(x))=limh0f(x+h)f(x)h=limh06(x+h)+2{6x+2}h=limh06x+6h+26x2h

Simplify the numerator and to obtain the derivatives f(x),

f(x) =limh06hh

Since the limit h approaches zero but not equal to zero, cancel the common term h from both the numerator and the denominator,

f(x)=limh0(6)=6

Thus, the second derivative of the function is, f(x)=6.

To check: The derivatives f(x), f(x),.and f(x) are reasonable by comparing the graphs of f(x), f(x). and f(x).

The derivative f(x), f(x). and f(x) are reasonable.

Graph:

Use the online graphing calculator to draw the graph of the functions f(x),f(x) and f(x) as shown below in Figure 1.

Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 2.7, Problem 45E

From Figure 1, it is observed that the graph of f(x) is a straight line and the graph of f(x) is a horizontal line.

Take several points on the domain and estimate the slope of the tangent to the function f(x) which is same as the point of f(x) at that point.

Thus, the derivative f(x) is reasonable.

The f(x) represents a linear function and f(x) represents a constant function.

The straight line has same slope at all points. That is, f(x) is constant value.

From the Figure 1, the estimated slope of the function f(x) is 6 which is same as the derivative of f(x). That is, f(x)=6.

Therefore, the function f(x) is reasonable.

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