## Derivative of Sine Functions

Question 1

Investigate the derivative of the function f(x) =sinx.

- Graph the function f(x) =sinx. For－2x2

ｙ＝f(x). Let f(x)=sinx ,－2x2

By sketching the graph we get:

Figure 1 : graph of function f(x) =sinx

Figure 1 reveals the range of the function f(x) =sinx. is [－1,1].

- Based on this graph, describe as carefully and fully as you can, the behabiour of the gradient of the function on the given domain.

The gradient of each point on the curve is valued as the gradient of the tangent line of the point.

According to the curve in figure 1, the behaviour of the gradient of the function indicates the following characteristics:

·The line of the tangent become flatter and flatter as the points move from left to right within

－2 to －,－ to －，０to ， to .

The line of the tangent become more and more precipitous as the points move from left to right within － to －，－ to 0， to ， to 2.

·In the domain of : ［－2，－［， ］－,［，］,2］the gradient is positive.

The gradient is negative in the domain of:：］－，－［, ］， ［.

·At the point when x equals to －,－,and ,the gradient is obviously 0.

c) Use your Graphics Calculator (GC) to find numerical values of the gradient of the function at every /4 unit. Sketch your findings on a graph.

- The numerical values (in 3 significant figures) of the gradient of the function at every /4 unit is shown in the table below:

2. The scatter plot (joined by a smooth curve) for the gradient of f(x) =sinx. at every /4 unit is shown below.

Gradient of f(x) =sinx.at every /4 unit

d) make a conjecture for the derived function f ′(x)

conjecture: f ′(x) = cosx

e) Use your GC to test your conjecture graphically. Explain your method and your findings. Modify your conjecture if necessary

1. The graph joined scatter plots with curve of f′(x) =cosx is shown in figure 2

The derivative of f(x) =sinx found by calculator with the curve in figure 2is shown in figure 3

Figure 2 scatter plots and f′(x) =cosx Figure 3 derivative of sine function found by calculator

Figure 2reveals that the curve of function f′(x) =cosx fits the scatter plots very well.

Figure 3reveals that the curve of the function found by the calculator overlaps exactly on the curve of f′(x) =cosx. Therefore the derivative of f(x) =sinx is: f (x) = cosx.

2. Method and findings of the derivative of f(x) =sinx:

By joining the scatter plots in figure 2 smoothly we can get a curve which indicates following

characteristics:

·symmetrical along x-axis.

·repeat its values every 2,(so the function is periodic with a period of 2)

·the maximum is 1 the minimum is -1

·the mean value of the function is zero

·the amplitude of the function is 1

Those are all characteristics of cosine function with the parameter of 1, according to the maximum and minimum values. Therefore the conjecture for the derived function f (x) is: f (x) = cosx

Question 2

Investigate the derivatives of functions of the form f (x)=asinx in similar way

a) Consider several different values of a.

Consider three values as a=－1,a=2 and a=－2 ,

1. When ① a=－1 f(x) =－sinx. ② a=－２ f(x) =－２sinx ③ a=2，f(x) =２sinx.

The curves of these functions are shown below:

graph of f(x)= －sinx f(x)=－2sinx , f(x)=2sinx

2. Description the gradient of the function:

The behaviour of the gradient of f(x) =２sinx. is the same as question 1(b)

The behaviour of the gradient of f(x) =－２sinx and f(x) =－sinx are opposite as question 1(b)

3. Values of gradient of the function at every /4 unit, sketch findings on a graph.

① when a=－1 f(x) =－sinx.

The value of the gradient of f(x) =－sinx. at every /4 unit is shown in the table:

The scatter plots of values in the table is sketched below: (joined by a smooth curve)

gradient of f(x) =－sinx.at every /4 unit

② when a=－２f(x) =－２sinx