The graph of f has a vertical tangent or a vertical cusp at c .

Single Variable Calculus: Concepts...

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

Single Variable Calculus: Concepts...

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

Solutions

Chapter 2.2, Problem 32E

(a)

To determine

To find:The graph of f has a vertical tangent or a vertical cusp at c .

Expert Solution

Explanation of Solution

Given:

The function f(x)=limx1x31x1

Concept used:

The definition of a vertical cusp is that the one sided limits of the derivative approach opposite ± : positive infinity on one side and negative infinity on the other side . A vertical tangent has the one sided limits of the derivative equal to the same sign of infinity

The derivative at the relevant point is undefined in both the cusp and the vertical tangent

Calculation:

The function is

f(x)=limx1x31x1............(1)

The vertical tangent means that the derivative at that point approaches infinity .

Since the slope is infinitely large .

Test one point in each of the region formed by the graph .

If the point satisfies the function then shade the entire region to denote that every point in the region satisfies the function .

To draw the table

f(x)=limx1x31x1

 x−axis 0 0.25 0.5 0.7 y−axis 1 2 3 4

To draw a graph

(b)

To determine

Expert Solution

Explanation of Solution

Given:

The function f(x)=limx1x31x1

Concept used:

The definition of a vertical cusp is that the one sided limits of the derivative approach opposite ± : positive infinity on one side and negative infinity on the other side . A vertical tangent has the one sided limits of the derivative equal to the same sign of infinity

The derivative at the relevant point is undefined in both the cusp and the vertical tangent

Calculation:

The function is

f(x)=limx1x31x1............(1)

The vertical tangent means that the derivative at that point approaches infinity .

Since the slope is infinitely large .

Test one point in each of the region formed by the graph .

If the point satisfies the function then shade the entire region to denote that every point in the region satisfies the function .

To draw the table

f(x)=limx1x31x1

 x−axis 0 0.25 0.5 0.7 y−axis 1 2 3 4

To draw a graph

Range of the x -axis is (0.5,1.5) to be with in a distance of 0.5 of 1

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