To find: The derivative of the function at if .
The value of is .
The given function is, .
The derivative of a function f at , denoted by , is
Difference of cube formula: .
Obtain the derivative of the function at .
Compute by using the equation (1),
Apply the difference of cube formula in the numerator as follows,
Simplify the denominator,
Thus, the value of is .
To Show: The function does not exist.
The derivative of a function f , denoted by , is
Consider the function .
Compute by using the equation (2),
Here, the function becomes larger and larger as h tends to zero. That is,
Therefore, the derivative of the function does not exist at .
Thus, the required proof is obtained.
To show: The has a vertical tangent line at
A curve has a vertical tangent line at if f is continuous at and
Consider the equation .
Substitute in ,
Thus is defined.
The limit of the function is computed as follows,
Therefore, is continuous at .
From part (a),
Take the limit of the function as x approaches zero.
Since the function is continuous at and .
By result, the curve has a vertical tangent at .
Thus, the curve has a vertical tangent at the point .
Use the online graphing calculator to draw the graph of the function as shown below in Figure 1,
From Figure 1, it is clear that the y-axis is the vertical tangent to the curve .
Subscribe to bartleby learn! Ask subject matter experts 30 homework questions each month. Plus, you’ll have access to millions of step-by-step textbook answers!