CalculusReview [328 marks]

Let f (x) = x3 − 2x2 + ax + 6. Part of the graph of f is shown in the following diagram.

The graph of f crosses the y-axis at the point P. The line L is tangent to the graph of f at P.

1a. Find the coordinates of P. [2 marks]

1b. Find f ′ (x). [2 marks]

1c. Hence, find the equation of L in terms of a. [4 marks]

1d. The graph of f has a local minimum at the point Q. The line L passes through Q. [8 marks]
Find the value of a.

6−2
 6−2x
2. Let f (x) = . The following diagram shows part of the graph of f . [8 marks]
√16+6x−x2

The region R is enclosed by the graph of f , the x-axis, and the y-axis. Find the area of R.

1 1
 1 1
Let f (x) = , for x > 2
.
√2x−1

3a. Find ∫ (f (x))2 dx. [3 marks]

Part of the graph of f is shown in the following diagram.

3b. Part of the graph of f is shown in the following diagram. [4 marks]

The shaded region R is enclosed by the graph of f, the x-axis, and the lines x = 1 and x = 9 .
Find the volume of the solid formed when R is revolved 360° about the x-axis.

Consider f(x), g(x) and h(x), for x∈R where h(x) = ( ∘ )(x).
4. Consider f(x), g(x) and h(x), for x∈R where h(x) = (f ∘ g)(x). [7 marks]
Given that g(3) = 7 , g′ (3) = 4 and f ′ (7) = −5 , find the gradient of the normal to the curve of h at
x = 3.

A function f (x) has derivative f ′ (x) = 3x2 + 18x. The graph of f has an x-intercept at x = −1.
 A function f (x) has derivative f ′ (x) = 3x2 + 18x. The graph of f has an x-intercept at x = −1.

5a. Find f (x). [6 marks]

5b. The graph of f has a point of inflexion at x = p. Find p. [4 marks]

Find the values of x for which the graph of f is concave-down.

5c. Find the values of x for which the graph of f is concave-down. [3 marks]

Consider a function f . The line L1 with equation y = 3x + 1 is a tangent to the graph of f when
x=2

6a. Write down f ′ (2). [2 marks]

6b. Find f (2). [2 marks]

2
 Let g (x) = f (x2 + 1) and P be the point on the graph of g where x = 1.

6c. Show that the graph of g has a gradient of 6 at P. [5 marks]

Let L be the tangent to the graph of g at P. L intersects L at the point Q.

6d. Let L2 be the tangent to the graph of g at P. L1 intersects L2 at the point Q. [7 marks]
Find the y-coordinate of Q.

A closed cylindrical can with radius r centimetres and height h centimetres has a volume of 20
 A closed cylindrical can with radius r centimetres and height h centimetres has a volume of 20 π
cm3.

7a. Express h in terms of r. [2 marks]

The material for the base and top of the can costs 10 cents per cm 2 and the material for the
 The material for the base and top of the can costs 10 cents per cm 2 and the material for the
curved side costs 8 cents per cm2. The total cost of the material, in cents, is C.

7b. Show that C = 20πr2 + 320π [4 marks]

r .

Given that there is a minimum value for C, find this minimum value in terms of .
7c. Given that there is a minimum value for C, find this minimum value in terms of π. [9 marks]

Let ( ) = 1 + e−x and ( ) = 2 + , for ∈ R, where is a constant.

 Let f(x) = 1 + e−x and g(x) = 2x + b, for x ∈ R, where b is a constant.

8a. Find (g ∘ f)(x). [2 marks]

8b. lim [4 marks]

Given that x→+∞(g ∘ f)(x) = −3, find the value of b.

( )= 2 − ∈R
 Let f(x) = x2 − x, for x ∈ R. The following diagram shows part of the graph of f .

The graph of f crosses the x-axis at the origin and at the point P(1, 0).

9a. Show that f ′ (1) = 1. [3 marks]

The line L is the normal to the graph of f at P.

9b. Find the equation of L in the form y = ax + b. [3 marks]

The line intersects the graph of at another point Q, as shown in the following diagram.
 The line L intersects the graph of f at another point Q, as shown in the following diagram.

9c. Find the x-coordinate of Q. [4 marks]

9d. Find the area of the region enclosed by the graph of f and the line L. [6 marks]

2−1
 2
10a. Find ∫ xex −1 dx. [4 marks]

2
10b. Find f(x), given that f ′ (x) = xex −1 and f(−1) = 3. [3 marks]

The following diagram shows the graph of ′, the derivative of .

 The following diagram shows the graph of f ′ , the derivative of f .

The graph of f ′ has a local minimum at A, a local maximum at B and passes through (4, − 2).

The point P(4, 3) lies on the graph of the function, f .

11a. Write down the gradient of the curve of f at P. [1 mark]

11b. Find the equation of the normal to the curve of f at P. [3 marks]

Determine the concavity of the graph of when 4 < < 5 and justify your answer.
11c. Determine the concavity of the graph of f when 4 < x < 5 and justify your answer. [2 marks]

A quadratic function f can be written in the form f(x) = a(x − p)(x − 3). The graph of f has
axis of symmetry x = 2.5 and y-intercept at (0, − 6)

12a. Find the value of p. [3 marks]

12b. Find the value of a. [3 marks]

The line = − 5 is a tangent to the curve of . Find the values of .

12c. The line y = kx − 5 is a tangent to the curve of f . Find the values of k. [8 marks]

( )= 2
 Let f(x) = x2 . The following diagram shows part of the graph of f .

The line L is the tangent to the graph of f at the point A(−k, k2 ), and intersects the x-axis at
point B. The point C is (−k, 0).

13a. Write down f ′ (x). [1 mark]

13b. Find the gradient of L. [2 marks]

13c. Show that the x-coordinate of B is − k . [5 marks]
2

13d. Find the area of triangle ABC, giving your answer in terms of k. [2 marks]

The region is enclosed by , the graph of , and the -axis. This is shown in the following
 The region R is enclosed by L, the graph of f , and the x-axis. This is shown in the following
diagram.

13e. Given that the area of triangle ABC is p times the area of R, find the value of p. [7 marks]
14. Let f ′ (x) = 3x2 [6 marks]
. Given that f(0) = 1, find f(x).
(x3+1)5

Let f(x) = cos x.

15a. (i) Find the first four derivatives of f(x). [4 marks]

(ii) Find f (19) (x).

( )= ∈ Z+
 Let g(x) = xk , where k ∈ Z+ .

15b. (i) Find the first three derivatives of g(x). [5 marks]

(ii) Given that g (19) (x) = k!
(xk−19 ), find p.
(k−p)!

Let k = 21 and h(x) = (f (19) (x) × g (19) (x)).

15c. (i) Find h′ (x). [7 marks]

−21! 2
(ii) Hence, show that h′ (π) = 2
π .

( )= 4 +5 ⩾ −1.25
 Let f(x) = √4x + 5, for x ⩾ −1.25.

16a. Find f ′ (1). [4 marks]

Consider another function g. Let R be a point on the graph of g. The x-coordinate of R is 1. The
equation of the tangent to the graph at R is y = 3x + 6.

16b. Write down g ′ (1). [2 marks]

16c. Find g(1). [2 marks]

( )= ( )× ( )
16d. Let h(x) = f(x) × g(x). Find the equation of the tangent to the graph of h at the [7 marks]
point where x = 1.

6−2x
Let f ′ (x) = 6x−x2
, for 0 < x < 6.

The graph of f has a maximum point at P.

17a. Find the x-coordinate of P. [3 marks]

The -coordinate of P is ln 27.

 The y-coordinate of P is ln 27.

17b. Find f(x), expressing your answer as a single logarithm. [8 marks]

17c. The graph of f is transformed by a vertical stretch with scale factor 1

ln 3
. The image of P under
this transformation has coordinates (a, b).
Find the value of a and of b, where a, b ∈ N.

⩽ ⩽
 The following diagram shows the graph of f(x) = 2x√a2 − x2 , for −1 ⩽ x ⩽ a, where a > 1
.

The line L is the tangent to the graph of f at the origin, O. The point P(a, b) lies on L.

18a. (i) 2a2−4x2 [6 marks]

Given that f ′ (x) = , for −1 ⩽ x < a, find the equation of L.
√a2−x2

(ii) Hence or otherwise, find an expression for b in terms of a.

The point Q( , 0) lies on the graph of . Let be the region enclosed by the graph of and
 The point Q(a, 0) lies on the graph of f . Let R be the region enclosed by the graph of f and
the x-axis. This information is shown in the following diagram.

Let AR be the area of the region R.

18b. Show that AR = 2 3 [6 marks]

3
a.

Let be the area of the triangle OPQ. Given that = , find the value of .
18c. Let AT be the area of the triangle OPQ. Given that AT = kAR , find the value of k. [4 marks]

Fred makes an open metal container in the shape of a cuboid, as shown in the following
diagram.

The container has height x m, width x m and length y m. The volume is 36 m3 .

Let A(x) be the outside surface area of the container.

108
19a. Show that A(x) =
x + 2x2 . [4 marks]

Find ′( ).
19b. Find A′ (x). [2 marks]

19c. Given that the outside surface area is a minimum, find the height of the container. [5 marks]

19d. Fred paints the outside of the container. A tin of paint covers a surface area of 10 m2 [5 marks]
and costs $20. Find the total cost of the tins needed to paint the container.

Let = ( ), for −0.5 ≤ x ≤ 6.5. The following diagram shows the graph of ′, the derivative
 Let y = f(x), for −0.5 ≤ x ≤ 6.5. The following diagram shows the graph of f ′ , the derivative
of f .

The graph of f ′ has a local maximum when x = 2, a local minimum when x = 4, and it
crosses the
x-axis at the point (5, 0).

20a. Explain why the graph of f has a local minimum when x = 5. [2 marks]

Find the set of values of for which the graph of is concave down.
20b. Find the set of values of x for which the graph of f is concave down. [2 marks]

20c. The following diagram shows the shaded regions A, B and C . [5 marks]

The regions are enclosed by the graph of f ′ , the x-axis, the y-axis, and the line x = 6.
The area of region A is 12, the area of region B is 6.75 and the area of region C is 6.75.
Given that f(0) = 14, find f(6).

The following diagram shows the shaded regions , and .

20d. The following diagram shows the shaded regions A, B and C . [6 marks]

The regions are enclosed by the graph of f ′ , the x-axis, the y-axis, and the line x = 6.
The area of region A is 12, the area of region B is 6.75 and the area of region C is 6.75.
2
Let g(x) = (f(x)) . Given that f ′ (6) = 16, find the equation of the tangent to the graph of g at
the point where x = 6.

′( )=3 2 −2 −9
 A function f has its derivative given by f ′ (x) = 3x2 − 2kx − 9, where k is a constant.

21a. Find f ′′ (x). [2 marks]

21b. The graph of f has a point of inflexion when x = 1. [3 marks]

Show that k = 3.

21c. Find f ′ (−2). [2 marks]

Find the equation of the tangent to the curve of at (−2, 1), giving your answer in
21d. Find the equation of the tangent to the curve of f at (−2, 1), giving your answer in [4 marks]
the form y = ax + b.

21e. Given that f ′ (−1) = 0, explain why the graph of f has a local maximum when [3 marks]
x = −1.

22. The following diagram shows the graph of f(x) = x

x2+1
, for 0 ≤ x ≤ 4, and the line [6 marks]
x = 4.

Let R be the region enclosed by the graph of f , the x-axis and the line x = 4.
Find the area of R.

Let
 Let
f(x) = px3 + px2 + qx.

23a. Find f ′ (x). [2 marks]

23b. Given that f ′ (x) ⩾ 0, show that p2 ⩽ 3pq. [5 marks]

Let
2x
f(x) = x2 +5
.

24a. Use the quotient rule to show that f ′ (x) = 10−2x2 [4 marks]
.
(x2+5)2

24b. Find ∫ 2x [4 marks]

x2+5
dx.

24c. The following diagram shows part of the graph of f . [7 marks]

The shaded region is enclosed by the graph of f , the x-axis, and the lines x = √5 and x = q.
This region has an area of ln 7. Find the value of q.

The following diagram shows part of the graph of

 The following diagram shows part of the graph of
y = f(x).

The graph has a local maximum at

A, where
x = −2, and a local minimum at
B, where
x = 6.

25a. On the following axes, sketch the graph of y = f ′ (x). [4 marks]

25b. Write down the following in order from least to greatest: f(0), f ′ (6), f ′′ (−2). [2 marks]

Consider the functions

f(x) ,
g(x) and
h(x) . The following table gives some valuesassociated with these functions.

26a. Write down the value of g(3) , of f ′ (3) , and of h′′ (2) . [3 marks]

The following diagram shows parts of the graphs of

 The following diagram shows parts of the graphs of
h and
h′′ .

There is a point of inflexion on the graph of

h at P, when
x=3.

26b. Explain why P is a point of inflexion. [2 marks]

Given that
h(x) = f(x) × g(x) ,

26c. find the y-coordinate of P. [2 marks]

26d. find the equation of the normal to the graph of h at P. [7 marks]

27. A rocket moving in a straight line has velocity v km s –1 and displacement s km at time [7 marks]
t seconds. The velocity v is given by v(t) = 6e2t + t . When t = 0 , s = 10 .
Find an expression for the displacement of the rocket in terms of t .

Let
f(x) = sin x + 12 x2 − 2x , for
0≤x≤π.

28a. Find f ′ (x) . [3 marks]

Let
g be a quadratic function such that
g(0) = 5 . The line
x = 2 is the axis of symmetry of the graph of
g.

28b. Find g(4) . [3 marks]

The function
 The function
g can be expressed in the form
g(x) = a(x − h)2 + 3 .

28c. (i) Write down the value of h . [4 marks]

(ii) Find the value of a .

28d. Find the value of x for which the tangent to the graph of f is parallel to the tangent to [6 marks]
the graph of g .

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