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3.05 Using the first and second derivatives

Worksheet
Graph original function from the gradient
1

The gradient function of g \left( x \right) is g' \left( x \right) = 3. Sketch a possible graph of g \left( x \right).

2

Sketch the linear function for which f \left( 0 \right) = 1 and f' \left( 2 \right) = 3.

3

Consider the graph of the gradient function f' \left( x \right):

What can be said about the graph of f \left(x\right)?

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f'(x)
4

The gradient function, f' \left( x \right) is graphed:

Sketch a possible graph of f \left( x \right).

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5

The gradient function, y' is graphed:

Sketch a possible graph of y.

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The gradient function, g' \left( x \right) is graphed:

Sketch a possible graph of g \left( x \right).

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7

Sketch a quadratic function, f \left( x \right), that satisfies the following conditions:

a
  • f \left( 0 \right) = - 18
  • f \left( 3 \right) = 0
  • f \left( 6 \right) = 6
  • f' \left( 6 \right) = 0
  • f' \left( x \right) > 0 for x < 6

b
  • f \left( 0 \right) = 16
  • f \left( 2 \right) = 0
  • f \left( 5 \right) = - 9
  • f' \left( 5 \right) = 0
  • f' \left( x \right) < 0 for x < 5

c
  • f \left( 0 \right) = 5
  • f \left( - 2 \right) = 0
  • f' \left( 3 \right) = 0
  • f' \left( x \right) > 0 for x < 3

d
  • f \left( 0 \right) = 10

  • f \left( - 2 \right) = 0

  • f \left( - 6 \right) = - 8

  • f' \left( - 6 \right) = 0

  • f' \left( x \right) < 0 for x < - 6

e
  • f \left( 0 \right) = - 3
  • f \left( - 3 \right) = 0
  • f \left( - 6 \right) = 1
  • f' \left( - 6 \right) = 0
  • f' \left( x \right) < 0 for x > - 6

8

Sketch a cubic function, f \left( x \right), that satisfies the following conditions:

a
  • f' \left( - 5 \right) = 0
  • f' \left( x \right) > 0 for all other values of x.

b
  • f' \left( 2 \right) = 0
  • f' \left( - 3 \right) = 0
  • f' \left( x \right) < 0 for - 3 < x < 2

  • f' \left( x \right) > 0 elsewhere

c
  • f \left( 0 \right) = 7
  • f \left( - 2 \right) = 0
  • f \left( - 4 \right) = - 1
  • f' \left( - 4 \right) = 0
  • f' \left( x \right) > 0 for x < - 4

  • f' \left( x \right) > 0 for x > - 4

9

Sketch a quartic function, f \left( x \right), that satisfies the following conditions:

a
  • f' \left( - 1 \right) = 0
  • f' \left( 4 \right) = 0
  • f' \left( x \right) > 0 for x > 4
  • f' \left( x \right) < 0 elsewhere
  • b
  • f \left( 0 \right) = 0
  • f' \left( 0 \right) = 0
  • f' \left( 2 \right) = 0
  • f' \left( - 2 \right) = 0
  • f' \left( x \right) > 0 for x < - 2, 0 < x < 2
  • f' \left( x \right) < 0 elsewhere
  • 10

    Sketch a function that matches the information for f \left( x \right):

    • f \left( 0 \right) = 0
    • f' \left( 0 \right) = 0
    • f' \left( - 2 \right) = 0
    • f' \left( 2 \right) = 0
    • f' \left( x \right) > 0 for - 2 < x < 2

    • f' \left( x \right) < 0 elsewhere

    The gradient function
    11

    Consider the gradient function f' \left( x \right) = 2 x + 4 graphed below:

    a

    State the x-intercept of the gradient function.

    b

    What feature of f \left( x \right) does this x-intercept represent?

    c

    Is the gradient of f \left( x \right) for x > - 2 positive or negative?

    d

    Is the gradient of f \left( x \right) for x < - 2 positive or negative?

    e

    What kind of turning point is at the point \left( - 2 , - 1 \right) on the graph of f \left( x \right)?

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    f'(x)
    12

    Consider the gradient function f' \left( x \right) = - 2 x + 8 graphed below:

    a

    State the x-intercept of the gradient function.

    b

    Is the gradient of f \left( x \right) for x > 4 positive or negative?

    c

    Is the gradient of f \left( x \right) for x < 4 positive or negative?

    d

    What kind of turning point is at the point \left(4, 17\right) on the graph of f \left( x \right)?

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    f'(x)
    13

    Consider the gradient function f' \left( x \right) = 3 x - 6.

    a

    Graph the gradient function.

    b

    State the x-intercept of the gradient function.

    c

    What kind of stationary point is at the point \left(2, - 5 \right) on the graph of f \left( x \right)?

    14

    Consider the gradient function f' \left( x \right) = - 4 x + 8.

    a

    Graph the gradient function.

    b

    State the x-intercept of the gradient function.

    c

    What kind of stationary point is at the point \left(2, 11\right) on the graph of f \left( x \right)?

    15

    Consider the graph of the gradient function \dfrac{d y}{d x} = 4:

    Find the equation of the function y if it has a y-intercept of 1.

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    \dfrac{dy}{dx}
    16

    Consider the gradient function \\ f' \left( x \right) = 2 \left(x + 3\right)^{2} graphed:

    a

    State the x-intercept of the gradient function.

    b

    Is the gradient of f \left( x \right) at values of x just to the left of - 3 positive or negative?

    c

    Is the gradient of f \left( x \right) at values of x just to the right of - 3 positive or negative?

    d

    What kind of feature is at the point \left( - 3 , 1\right) on the graph of f \left( x \right)?

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    f'(x)
    17

    Consider the gradient function \\ f' \left( x \right) = 3 \left(x - 6\right)^{2} graphed:

    a

    State the x-intercept(s) of the gradient function.

    b

    What kind of feature is at the point \left(6, 2\right) on the graph of f \left( x \right)?

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    f'(x)
    18

    Consider the gradient function \\ f' \left( x \right) = - 6 \left(x + 5\right)^{2} graphed:

    a

    State the x-intercept(s) of the gradient function.

    b

    What kind of feature is at the point \left( - 5 , 3\right) on the graph of f \left( x \right)?

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    y
    19

    Consider the gradient function \\ f' \left( x \right) = - 5 \left(x - 2\right)^{2} graphed:

    a

    State the x-intercept of the gradient function.

    b

    What kind of feature is at the point \left(2, 1\right) on the graph of f \left( x \right)?

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    20

    The function f \left( x \right) has a derivative given by f' \left( x \right) = 6 \left(x + 7\right) \left(x + 2\right). A graph of the derivative function is shown:

    a

    State the x-intercept(s) of the gradient function.

    b

    What kind of feature is at the point \left( - 7 , 52\right) on the graph of f \left( x \right)?

    c

    What kind of feature is at the point \left( - 2 , - 73 \right) on the graph of f \left( x \right)?

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    f'(x)
    21

    Consider the gradient function \\ f' \left( x \right) = - 6 \left(x - 1\right) \left(x - 5\right) graphed:

    a

    State the x-intercept(s) of the gradient function.

    b

    What kind of feature is at the point \left(1, - 13 \right) on the graph of f \left( x \right)?

    c

    What kind of feature is at the point \left(5, 51\right) on the graph of f \left( x \right)?

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    f'(x)
    22

    Consider the gradient function \\ f' \left( x \right) = \left(x - 2\right)^{2} - 4 graphed:

    a

    State the coordinates of the turning point of f' \left( x \right).

    b

    What kind of feature is at the point \left(2, - 5 \right) on the graph of f \left( x \right)?

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    f'(x)
    23

    Consider the gradient function f' \left( x \right) = - \left(x - 1\right)^{2} + 2.

    a

    Graph the gradient function.

    b

    State the coordinates of the turning point of f' \left( x \right).

    c

    What kind of feature is at the point \left(1, 4\right) on the graph of f \left( x \right)?

    24

    Consider the gradient function f' \left( x \right) = 6 \left(x + 6\right) \left(x + 2\right).

    a

    Graph the gradient function.

    b

    What kind of feature is at the point \left( - 6 , 2\right) on the graph of f \left( x \right)?

    c

    What kind of feature is at the point \left( - 2 , - 62 \right) on the graph of f \left( x \right)?

    25

    Consider the gradient function f' \left( x \right) = \left(x + 7\right)^{2} - 9.

    a

    Graph the gradient function.

    b

    State the coordinates of the turning point on the graph of the gradient function.

    c

    What kind of feature is at the point \left( - 7 , 64\right) on the graph of f \left( x \right)?

    d

    What kind of feature is at the point \left( - 10 , 82\right) on the graph of f \left( x \right)?

    e

    What kind of feature is at the point \left( - 4 , 46\right) on the graph of f \left( x \right)?

    26

    Consider the gradient function f' \left( x \right) = - \left(x - 3\right)^{2} + 9.

    a

    Graph the gradient function.

    b

    State the coordinates of the turning point on the graph of the gradient function.

    c

    What kind of feature is at the point \left(3, 30\right) on the graph of f \left( x \right)?

    d

    What kind of feature is at the point \left(0, 12\right) on the graph of f \left( x \right)?

    e

    What kind of feature is at the point \left(6, 48\right) on the graph of f \left( x \right)?

    27

    Consider the gradient function f' \left( x \right) = 12 \left(x + 2\right)^{2} \left(x + 5\right).

    a

    Graph the gradient function.

    b

    What kind of feature is at the point \left( - 5 , - 225 \right) on the graph of f \left( x \right)?

    c

    What kind of feature is at the point \left( - 2 , - 144 \right) on the graph of f \left( x \right)?

    28

    Consider the gradient function f' \left( x \right) = 12 \left(x + 4\right) \left(x + 1\right) \left(x - 4\right):

    a

    What kind of feature is at the point \left( - 4 , - 256 \right) on the graph of f \left( x \right)?

    b

    What kind of feature is at the point \left( - 1 , 95\right) on the graph of f \left( x \right)?

    c

    What kind of feature is at the point \left(4, - 1280 \right) on the graph of f \left( x \right)?

    d

    What kind of feature is at the point \left(2, - 688 \right) on the graph of f \left( x \right)?

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    y
    Graphs of derivative functions
    29

    Consider the following graph of y = f \left( x \right):

    Sketch the graphs of y = f' \left( x \right) and \\ y = f'' \left( x \right).

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    30

    Consider the following graph of y = f' \left( x \right):

    a

    Sketch the graph of y=f''(x).

    b

    For what intervals is the graph of \\ y = f \left( x \right) increasing?

    c

    For what interval is the graph of \\ y = f \left( x \right) concave up?

    d

    What are the x-values of the stationary points of y = f \left( x \right)?

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    31

    Draw the gradient function for the following graphs:

    a
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    b
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    c
    -7
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    d
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    e
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    y
    f
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    y
    32

    The diagram shows the graph of y = f \left( x \right):

    a

    State the interval(s) where the values of the derivative f' \left( x \right) are negative.

    b

    What happens to f' \left( x \right) for large values of x?

    c

    Draw the graph of y = f' \left( x \right).

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    ACMMM112

    sketch the graph of a function using first and second derivatives to locate stationary points and points of inflection

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    identify families of curves with the same derivative function

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