1.

A function is given as f(x)=3x2−6x+4+3x,−2≤x≤4,x≠0f(x)=3x2−6x+4+3x,

2.
Consider the graph of the function f(x)=7−3x2−x3f(x)=7−3x2−x3

(a) Label the local maximum as A on the graph.

(b) Label the local minimum as B on the graph.

(c) Write down the interval where f(x)>5f(x)>5.

3.
The graph of f(x)=8−x2f(x)=8−x2 crosses the x-axis at the points A and B.

(a) Find the x-coordinate of A and of B.

(b) The region enclosed by the graph of ff and the x-axis is revolved 360o about the x-axis. Find
the volume of the solid formed.

4.
The acceleration, aa ms-2 , of an object at time tt seconds is given by
 a=1t+4sin3t,(t≥1)a=1t+4sin3t,(t≥1)
The object is at rest when t=1t=1.
Find the velocity of the object when t=7t=7.

5.
Let f(x)=2xx2+3f(x)=2xx2+3
(a) Use the quotient rule to show that f′(x)=6−2x2(x2+3)2f′(x)=6−2x2(x2+3)2.
(b) Find ∫2xx2+3dx∫2xx2+3dx.

(c) The area enclosed by the graph of f(x)f(x), the x-axis and the lines x=3–
√x=3 and x=nx=n has an area of ln14ln⁡14. Find the value of n.

6.
(a) Find dydxdydx when:
y=(7−5x2)12y=(7−5x2)12
(b) Find the following integral:

∫(1−cos3x)dx∫(1−cos3x)dx

Worked Solution

7.
Consider the function f(x)=6−ax+3x2,x≠0f(x)=6−ax+3x2,x≠0
(a) Write down the equation of the vertical asymptote of the graph of y=f(x)y=f(x).
(b) Write down f′(x)f′(x)
The line T is the tangent to the graph of y=f(x)y=f(x) at the point where x=1x=1 and it has a
gradient of -8.
(c) Show that a=2a=2.
(d) Find the equation of T.

(e) Using your calculator find the coordinates of the point where the graph
of y=f(x)y=f(x) intersects the x-axis.
(f) The line T also intersects f(x)f(x) when −2≤x≤0−2≤x≤0. Find the coordinates of this
intersection.
Worked Solution

8.
If f(x)=xsinxf(x)=xsinx, for −3≤x≤3−3≤x≤3
(a) Find f′(x)f′(x)
(b) Sketch the graph of y=f′(x)y=f′(x), for −3≤x≤3−3≤x≤3
Worked Solution

9.
Consider the cubic function f(x)=16x3−2x2+6x−2f(x)=16x3−2x2+6x−2
(a) Find f′(x)f′(x)
The graph of ff has horizontal tangents at the points
where x=ax=a and x=bx=b where a<ba<b.
(b) Find the value of aa and the value of bb
(c) Sketch the graph of y=f′(x)y=f′(x).
(d) Hence explain why the graph of ff has a local maximum point at x=ax=a.
(e) Find f′′(b)f″(b).
(f) Hence, use your answer to part (e) to show that the graph of ff has a local minimum point
at x=bx=b.
(g) Find the coordinates of the point where the normal to the graph of ff at x=ax=a and the
tangent to the graph of ff at x=bx=b intersect.
Worked Solution

10.
The following diagram shows part of the graph of:

f(x)=(9+4x)(6−x)10,x∈Rf(x)=(9+4x)(6−x)10,x∈R
The shaded region A is bounded by the x-axis, y-axis and the graph of ff.
(a) Write down an integral for the area of region A.

(b) Find the area of region A.

The three points A (0 , 0) , B (6 , 9) and C (p , 0) define the vertices of a triangle.

Find the value of p, the x-coordinate of C , such that the area of the triangle is equal to the area of
region A.

Worked Solution

11.
Consider the function f(x)=x3−9x+2f(x)=x3−9x+2.
(a) Sketch the graph
of y=f(x)y=f(x) for −4≤x≤4−4≤x≤4 and −14≤y≤14−14≤y≤14 showing clearly the
axes intercepts and local maximum and minimum points. Use a scale of 2 cm to represent 1 unit
on the x-axis, and a scale of 1 cm to represent 2 units on the y-axis.
(b) Find the value of f(−1)f(−1).
(c) Write down the coordinates of the y-intercept of the graph of f(x)f(x).
(d) Find f′(x)f′(x).
(e) Find f′(−1)f′(−1)
(f) Explain what f′(−1)f′(−1) represents.
(g) Find the equation of the tangent to the graph of f(x)f(x) at the point where x is –1.
R and S are points on the curve such that the tangents to the curve at these points are horizontal.
The x-coordinate of R is aa , and the x-coordinate of S is bb , b>ab>a.
(h) Write down the value of aa ;
(i) Write down the value of bb.
(j) Describe the behaviour of f(x)f(x) for a<x<ba<x<b.
Worked Solution

12.
Consider the graph of the function f(x)=x2+2f(x)=x2+2.
(a) Find the area between the graph of ff and the x-axis for 2≤x≤32≤x≤3.

(b) If the area described above is rotated 360o around the x-axis find the volume of the solid
formed.

Worked Solution

13.
Let f(x)=g(x)h(x)f(x)=g(x)h(x), where g(3)=36g(3)=36, h(3)=12h(3)=12, g′(3)=10g
′(3)=10 and h′(3)=4h′(3)=4. Find the equation of the normal to the graph of ff at x=3x=3.
Worked Solution

14.
(a) Express the algebraic fraction

6x2−47x+49(5−x)(1−2x)6x2−47x+49(5−x)(1−2x)
in the form

A+B5−x+C1−2xA+B5−x+C1−2x
where AA, BB and CC are integers.

(b) Hence show that the following integral equates to 3.03 correct to three significant figures.

∫0.2506x2−47x+49(5−x)(1−2x)dx∫00.256x2−47x+49(5−x)(1−2x)dx

Worked Solution

15.
The displacement, in millimetres, of a particle from an origin, O, at time t seconds, is given
by s(t)=t3cost+5tsints(t)=t3cost+5tsint where 0≤t≤50≤t≤5 .

(a) Find the maximum distance of the particle from O.

(b) Find the acceleration of the particle at the instant it first changes direction.

Worked Solution
16.
The function ff is defined as follows:
f(x)=1221+60e−0.3xf(x)=1221+60e−0.3x
(a) Calculate f(0)f(0).
(b) Find a value of xx for which f(x)=85f(x)=85
(c) Find the range of ff.

(d) Show that:

f′(x)=2196e−0.3x(1+60e−0.3x)2f′(x)=2196e−0.3x(1+60e−0.3x)2
(e) Find the maximum rate of change of ff.
Worked Solution

17.
The following diagram shows part of the graph of y=f(x)y=f(x)

The graph has a local maximum where x=−23x=−23, and a local minimum where x=4x=4.
sketch the graph of y=f′(x)y=f′(x) for −4≤x≤7−4≤x≤7
Write down the following in order from least to greatest: f(2),f′(4)f(2),f′(4) and f′′(4)f″(4).
Worked Solution

18.
Make a sketch of a graph showing the velocity (in ms−1ms−1) against time of a particle
travelling for six seconds according to the equation:
 v=esint−1v=esin⁡t−1
(a) Find the point at which the graph crosses the tt axis.

(b) How far does the particle travel during these first six seconds?

Worked Solution

19.
Find the value of aa if π<a<2ππ<a<2π and:

∫aπsin3xdx=−13∫πasin3xdx=−13

Worked Solution

20.

This graph represents the function f:x→acosx,a∈Nf:x→acos⁡x,a∈N

(b) Find the area of the shaded region.

Worked Solution

21.
Find f(x)f(x) if f′(x)=6sin2xf′(x)=6sin⁡2x and the graph of f(x)f(x) passes through the
point (π3,11)(π3,11).
 Worked Solution

22.
A particle P moves along a straight line. The velocity vv in metres per second of P
after tt seconds is given by v(t)=3sint−8tcost,0≤t≤7v(t)=3sin⁡t−8tcos⁡t,0≤t≤7.

(a) Find the initial velocity of P.

(b) Find the maximum speed of P.

(c) Write down the number of times that the acceleration of P is 0 ms-2.

(d) Find the acceleration of P at a time of 5 seconds.

(e) Find the total distance travelled by P.

Worked Solution

23.
Let f(x)=ln3xkxf(x)=ln3xkx where x>0x>0 and k∈Q+k∈Q+.
(a) Find an expression for the first derivative f′(x)f′(x).
The graph of ff has exactly one maximum point at P.

(b) Find the x-coordinate of P.

The graph of ff has exactly one point of inflection at Q.

(c) Find the x-coordinate of Q.

(d) The region enclosed by the graph of ff, the x-axis, and the vertical lines through P and Q has
an area of one square unit, find the value of kk.
Worked Solution

24.
The diagram shows a sketch of the curve C with equation:
 y=2x32−4x+3y=2x32−4x+3
The point P with coordinates (4, 3) lies on C.

The line L is the tangent to C at the point P.

The region A, shown shaded, is bounded by the curve C, the line L and the y-axis.

Find the area of A making your method clear.

Note that solutions based entirely on graphical or numerical methods are not acceptable.)

Worked Solution

25.
The following diagram shows the graph
of f(x)=cos(ex)for0≤x≤0.5f(x)=cos⁡(ex)for0≤x≤0.5.
(a) Find the x-intercept of the graph of f(x)f(x).
The region enclosed by the graph of f(x)f(x), the y-axis and the x-axis is rotated 360° about the
x-axis.

(b) Find the volume of the solid formed.

Worked Solution

26.
The following equation defines a curve which passes through A(2π,3π)A(2π,3π)
sin2y+ycos22x=x+csin⁡2y+ycos2⁡2x=x+c
(a) Find the exact value of the constant cc.
(b) Find an expression for dydxdydx
(c) Find the equation of the normal to the curve at A, giving your answer in the
form y=mx+by=mx+b
Worked Solution

27.
Find f(x)f(x) if:
f′(x)=12x3x2+4−−−−−−√f′(x)=12x3x2+4
given that f(0)=10f(0)=10
Worked Solution

28.
Consider the function f(x)=20x2+kxf(x)=20x2+kx where kk is a constant and x≠0x≠0.
(a) Write down f′(x)f′(x)
The graph of y=f(x)y=f(x) has a local minimum point at x=2x=2.
(b) Show that k=5k=5.
(c) Find f(1)f(1).
(d) Find f′(1)f′(1).
(e) Find the equation of the normal to the graph of y=f(x)y=f(x) at the point where x=1x=1
     Give your answer in the form ay+bx+c=0ay+bx+c=0 where a,b,c∈Za,b,c∈Z
(f) Sketch the graph of y=f(x)y=f(x) ,
for −5≤x≤10−5≤x≤10 and −10≤y≤50−10≤y≤50.
(g) Write down the coordinates of the point where the graph of y=f(x)y=f(x) intersects the x-
axis.
(h) State the values of xx for which f(x)f(x) is decreasing.