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3.04 Graphing techniques using calculus

Worksheet
Features of graphs
1

State the type of point that matches the following descriptions:

a

A point where the curve changes from decreasing to increasing.

b

A point where the curve changes from increasing to decreasing.

c

A point where the tangent is horizontal and the concavity changes about the point.

2

Consider the function y = e^{x} \left(x - 3\right).

a

Find the coordinates of the turning point.

b

State whether this is a minimum or maximum turning point.

3

Consider the function f \left( x \right) = 2 x^{3} - 12 x^{2} + 18 x + 3.

a

Find the x-coordinates of the turning points.

b

State the coordinates of the local maximum.

c

State the coordinates of the local minimum.

d

State the absolute maximum value on [0,7].

e

State the absolute minimum value on [0,7].

4

Consider the function f \left( x \right) = \dfrac{4 x^{9}}{\left(x + 2\right)^{4}}.

a

Find f' \left( 2 \right).

b

Is the function increasing or decreasing at x = 2?

5

Consider the function y = \dfrac{\sin x}{1 + \cos x}.

a

Find \dfrac{dy}{dx}.

b

Determine the number of turning points function y has.

Graphs of functions
6

For each of the following functions:

i

Find the y-intercept.

ii

Find the x-intercepts.

iii

Find f' \left( x \right).

iv

Hence find the x-coordinates of the stationary points.

v

Classify the stationary points.

vi

Sketch the graph of the function.

a
f \left( x \right) = 9 x^{2} + 18 x - 16
b
f \left( x \right) = \left( 4 x + 5\right)^{2} \left(x - 1\right)
c
f \left( x \right) = \left( 2 x - 1\right)^{2} \left(1 - x\right)
d
f \left( x \right) = \left(x + 2\right)^{3}-1
e
f \left( x \right) = x^{3} + 11 x^{2} + 24 x
f
f \left( x \right) = \left(x^{2} - 4\right)^{2} + 4
g
f(x)=2\left(x-1\right)^3 - 16
h
f(x)=x^4 - 4x^3
i
f(x) = x^3 + 5x^2
j
f(x)=x^2(x-3)^2
k
f \left( x \right) = \left( 3 x - 2\right) \left(x + 3\right)
7

Consider the equation of the parabola y = 3 x^{2} - 18 x + 24.

a

Find the x-intercepts.

b

Find the y-intercept.

c

Find \dfrac{dy}{dx}.

d

Find the stationary point.

e

Classify the stationary point.

f

Sketch the graph of the parabola.

8

Consider the curve f \left( x \right) = x^{3} - 5 x^{2} + 3 x - 5.

a

Find the x-coordinate(s) of the turning point(s).

b

Determine an equation for f'' \left( x \right).

c

Classify both stationary points

d

Find the x-coordinates of the possible point of inflection, and verify that it is a point of inflection.

e

Sketch the graph of f \left( x \right) = x^{3} - 5 x^{2} + 3 x - 5.

9

Consider the curve y = x^{3} - 6 x^{2} - 3.

a

Find the coordinates of the stationary points.

b

Classify the stationary points.

c

Find the x-coordinates of the possible point of inflection, and verify that it is a point of inflection.

d

Sketch the graph of y = x^{3} + 6 x^{2} - 3.

10

Consider the function f \left( x \right) = \left(2 - 3 x\right)^{3} + 8.

a

State the coordinates of the y-intercept.

b

Find the the x-intercept(s).

c

Determine an equation for f' \left( x \right).

d

Hence find the x-coordinate(s) of the stationary point(s).

e

Classify the stationary point(s).

f

Sketch the graph of the function.

11

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

a

Find the coordinates of the turning points.

b

Classify the stationary points.

c

Find the x-coordinate(s) of any potential point(s) of inflection, and verify that they are point(s) of inflection.

d

Sketch the graph of f \left( x \right).

Limiting behaviours of functions
12

Consider the function f \left( x \right) = 4 e^{ - x^{2} }.

a

Find f' \left( x \right).

b

Find the values of x for which:

i
f' \left( x \right) = 0
ii

f' \left( x \right) \gt 0

iii

f' \left( x \right) \lt 0

c

Find the limit of f(x) as x \to \infty.

d

Find the limit of f(x) as x \to - \infty.

e

Sketch the graph of f \left( x \right).

13

Consider the function f \left( t \right) = \dfrac{4}{2 + 3 e^{ - t }}.

a

Find f \left( t \right) when t = 0.

b

Find f' \left( t \right).

c

State whether f(x) is an increasing or decreasing function. Explain your answer.

d

Hence, state how many stationary points the function has.

e

Find the limit of f (t) as t \to \infty.

14

The graph of the function f \left( x \right) = e^{ 2 x} \sin 3 x is shown:

a

Find f' \left( x \right).

b

Find the x-intercepts, B and C.

c

Determine the coordinates of point A, correct to two decimal places.

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Outcomes

ACMMM110

understand the concepts of concavity and points of inflection and their relationship with the second derivative

ACMMM111

understand and use the second derivative test for finding local maxima and minima

ACMMM112

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

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