f (x) = sin x , What is the y-intercept of the graph of f?

Question

f (x) = sin x

(a) What is the y-intercept of the graph of f?

(b) For what numbers x, - π ≤ x ≤ π, is the graph of f increasing?

(c) What is the absolute maximum of f?

(d) For what numbers x, 0 ≤ x ≤ 2π,  does f (x) = 0?

(e) For what numbers x, - 2π ≤ x ≤ 2π, does f (x) = 1? Where does f (x) = - 1?

(f) For what numbers x, - 2π ≤ x ≤ 2π, does f (x) = -1/2?

(g) What are the x-intercepts of f ?

Answer 1

Step 1 :

(a)

The function .

Find the y - intercept :

Let the function

Find the y - intercept, by substituting in .

.

The y - intercept is zero.

Step 2 :

(b)

The function and the interval is .

Differentiate the function with respect to x.

Determination of critical points :

The critical points exist when .

Equate to zero:

Solve in the interval .

General solution of is , where n is an integer.

General solution is .

If , .

If , .

The solutions are in the interval .

Comments

Contd......

The critical points are and the test intervals are.

Interval	Test Value	Sign of	Conclusion
			Decreasing
			Increasing
			Decreasing

The function is increasing over the interval .

Solution :

(a) The y - intercept is zero.

(b) The function is increasing over the interval .

Answer 2

Step 1 :

(c)

The function and the interval is .

The critical points are .

Find the values of at these critical points.

.

.

Find the values of at the end points of the interval.

.

.

Compare the four values of to find the absolute maximum.

Absolute maximum value is .

Step 2 :

(d)

The function and the interval is .

General solution of is , where is an integer.

General solution : .

If , .

If , .

If , .

The solutions are in the interval .

Solution :

(c) Absolute maximum value is .

(d) The solutions are in the interval .

Answer 3

Step 1 :

(e)

The function and the interval is .

General solution of is , where is an integer.

General solution : .

If , .

If , .

If , .

The solution is in the interval .

General solution of is , where is an integer.

General solution : .

If , .

If , .

If , .

If , .

The solutions are in the interval .

Solution :

The solutions are , in the interval .

Answer 4

Step 1 :

(f)

General solution of is , where is an integer.

General solution : .

If , .

If , .

If , .

If , .

The solutions are in the interval .

Step 2 :

(g)

 Find the x - intercept :

The function .

Let the function .

Find the x - intercept, by substituting in .

General solution of is , where is an integer.

General solution is , where is an integer.

The x - intercepts are , where is an integer.

Solution :

(f) The solutions are in the interval .

(g) The x - intercepts are , where is an integer.