help? derivative-trig

Question

Answer 1

(7)

Step 1:

The function is and .

Slope of the tangent line equation is derivative of the function.

.

Differentiate on each side with respect to .

.

Product rule of derivatives : .

Substitute in derivative function.

Slope of the tangent line equation is .

Step 2:

Find the point of tangency.

Substitute in the function.

Tangent point is .

Find the equation of tangent line.

Point slope form of the line equation : .

Substitute and in point slope form of line equation.

Tangent line equation is .

Solution :

Tangent line equation is .

Answer 2

(8)

Step 1:

The function is .

Differentiate on each side with respect to .

.

Find the critical points.

A critical number of a function is a number in the domain of such that either or does not exist.

Find the critical point by graphing the derivative function.

Graph:

Graph the derivative function .

Observe the graph:

Locate the values of x where the value of the derivative function is zero.

Critical points are and .

Comments

Continued...

Step 2:

The range is .

The test intervals are , , , and .

Therefore the function is increasing on the intervals , and .

The function is decreasing on the intervals and .

Solution:

The function is increasing on the intervals , and  .

The function is decreasing on the intervals and  .