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6SECTION GRAPHS OF DERIVATIVES

37

We must be able to recognise and sketch the graph of the derivatives of functions.

Steps in graphing the derivative of a function• The turning points of graphs are very important. At a turning point, the graph of the derivative

will cross the x-axis. Plot these points first.• Then look for where the given graph is increasing or decreasing:

° When the graph is increasing, its derivative will be positive and so it will be above the x-axis.

° When the graph is decreasing, its derivative will be negative and so it will be below the x-axis.

Look at the next example for a step-by-step solution for graphing the derivative of a function.

When we differentiate a cubic function, we get a quadratic function.When we differentiate a quadratic function, we get a linear function.

When we differentiate a linear function, we get a horizontal line.

The diagram shows the graph of a cubic function, g (x).

On the same axis and scale, sketch the graph of:

(i) g!(x)

(ii) g"(x)

(iii) g!"(x), where g "!(x) is the third derivative of g(x)

Solution(i) g!(x)

Since g(x) is a cubic function, its derivative, g!(x), is a quadratic function.• At turning points of g(x), g!(x) # 0.• Therefore, at the turning point of g(x), the graph of its derivative, g!(x), will cut the

x-axis.• Between the turning points, the graph of the cubic function is decreasing, therefore

the graph of g!(x) will be below the x-axis between these two points.

g(x)

x

y

NCPM_5_supplement_SEC6.qxd 8/28/14 2:11 PM Page 37

38

NEW CONCISE PROJECT MATHS 5

• Outside of the turning points, the graph of the cubic function is increasing,therefore the graph of g!(x) will be above the x-axis between these two points.

Putting all of this together gives:

(ii) g"(x)At the turning point of g!(x), g"(x) # 0.

• Therefore, at the turning point of g!(x), the graph of its derivative, g"(x), will cutthe x-axis.

• To the left of the turning points, the graph of the quadratic function, g!(x), isdecreasing, therefore the graph of g"(x) will be below the x-axis.

• To the right of the turning points, the graph of the quadratic function, g!(x), isincreasing, therefore the graph of g"(x) will be above the x-axis.

Putting all of this together gives:

g!(x)

g!!(x)

g(x)

x

y

g!(x)g(x)

x

y


(iii) g!"(x)The graph of g"(x) is a straight line with a positive slope and therefore it does nothave any turning points. It is increasing at a constant rate for all values of x, thereforeits derivative will be a constant positive value.Hence, any horizontal line above the x-axis could represent g!"(x).

g!(x)

g!!(x)

g!!!(x)

g(x)

x

y

GRAPHS OF DERIVATIVES

39

Exercise 6.11. The diagrams show the graph of a quadratic function, g(x).

Copy the sketch of each graph into your copy book. On the same axis and scales sketch the graph of:

(a) g!(x), the first derivative of g(x) (b) g"(x), the second derivative of g(x)

(i) (ii)

g(x)

x

y

g(x)

x

y

NCPM_5_supplement_SEC6.qxd 8/28/14 2:11 PM 9

40


2. Give three reasons why the given line represents the slope function of the given curve.

(i) (ii)

3. The diagram shows the graph of a cubic function, g(x).Copy the sketch of the graph into yourcopybook. On the same axis and scale, sketch the graph of:

(i) g!(x)

(ii) g"(x)

(iii) g!"(x)

4. The graphs of the slope functions of two curves are given below. Find:

(a) The x-coordinate of the stationary points of each curve

(b) The range of values of x for which each curve is increasing

(i) (ii) y

1 2 3 x−3 −2 −1 0−4−5

f 9(x) = x2 + 2x – 3y

1 2 3 x−3 −2 −1 0 4 5−4−5−6−7

f 9(x) = –x2 – 3x + 10

y

x

g(x)

y

2 x

y

2 x

NCPM_5_supplement_SEC6.qxd 8/28/14 2:11 PM 0

GRAPHS OF DERIVATIVES

41

5. The diagram shows the graph of a function, f(x).

(i) Identify the function f (x).

(ii) Find f !(x), the first derivative of f (x).

(iii) On the same axis and scale, graph f !(x), the first derivative of f (x).

(iv) Find f "(x), the second derivative of f (x).

(v) On the same axis and scale, graph f "(x), the second derivative of f (x).

6. Each diagram below shows part of the graph of a function. Each of these functions is either

quadratic, cubic, trigonometric or exponential (not necessarily in that order).

Each diagram below shows part of the graph of the first derivative of one of the abovefunctions (not necessarily in the same order).

A B C D

f g h k

1

y

x0

0

−1

π 2π 3π 4π2π

23π

25π

27π

f(x)


3. g(x) ! 3f(x)

h(x) ! f(x + 2) + 2

4.

5. Graph B: y ! x2 + 3

Graph C: y ! (x − 2)2

Graph D: y ! (x + 3)2 − 2

Section 5 – Continuity of Functions

1. x = −1 2. and 3. x = 1 4. x = −3 5. x = −2

6. x = −5 and x = 2

Section 6 – Graphs of Derivatives

1. (i) (ii) y

x

y

g(x)

g''(x)

g'(x)

y

x

g(x)

g!!(x)

g!(x)

x = 3p2

x = p2

y

1

2

3

1 2 x−3 −2 −10

−1

−2

−3

f(x) + 2f(x)

f(x + 2) – 1

46


NCPM_5_supplement_ANS copy.qxd 8/28/14 2:06 PM Page 46

47

ANSWERS

2. Curve is decreasing for x < 2, therefore the slope is negative.

Turning point at x = 2, therefore the slope is zero when x = 2.

Curve is increasing for x > 2, therefore the slope is positive.

3. 4. (i) (a) x = −5, 2 (b) −5 < x < 2

(ii) (a) x = −3, 1 (b) x < −3, x > 1

5. (i) f(x) = sin x (ii) f "(x) = cos x

(iii) (iv) f #(x) = −sin x

(v)

0

1

–1

02π π 2π 3π 4π

23π

25π

27π

f '(x) = cos x

f ''(x) = –sin x

f (x) = sin x

y

x

0

1

–1

02π π 2π 3π 4π

23π

25π

27π

f '(x) = cos x

f (x) = sin x

y

x

x

y

g(x)

g!!(x)

g!(x)

g!!!(x)


6. (i)

(ii) For the quadratic function:

The first derivative will be a straight line.

The second derivative will be a horizontal line.

For the cubic function:

The first derivative will be a quadratic line.

The second derivative will be a straight line.

For the trigonometric function:

The given function is the sine function.

The first derivative will be the cosine function.

The second derivative will be −1(sine function), so it will be the original function upside-down.

For the exponential function:

The given function is an exponential function with a negative power, hence it is decreasing.

The first derivative will be the −1(the exponential function), so it will be similar to the original

function, but upside-down.

The second derivative will be the −1(the first derivative), so it will be back to the same

orientation as the original function.

Type of function Function 1st derivative 2nd derivative

Quadratic k B I

Cubic f D II

Trigonometric g A III

Exponential h C IV

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