Question 1
Express the quadratic
f (x) = 4x2 8x 13
in the form a(x A)2 + B and hence sketch its graph.
Solution
f (x) = 4x2 8x 13
= 4 x2 + 2x 13
xunit=0.7cm,yunit=0.08cm
[Oy=0,Dy=9](0,0)(2.5,-34)(-4,0)
= 4 (x + 1)2 1 13
dup mul mul 9 sub *(-1,-9).1
= 4(x
Question 1
The current, i(t), is given by the formula
i(t) =
mA,
for t 0.
Plot the graph of ln(i(t) against t.
Solution
Since
exp(i(t) = ln(3) 4t
we want the graph of y = ln(3) 4t
xunit=1.2cm,yunit=.25cm (-1,-10)(5,5)
[dx=0,Dx=2,Dy=5]-(0,0)(-0.5,-10)(5,5)
Question 1
State the domain and range of the function
f (x) = 3 + e2x .
Hence sketch the graph of y = f (x).
Solution
Domain of f (x)
Division by zero
never occurs.
Square root of negative number
never occurs.
Log of non-positive number
never occurs
Question 1
Use the limit denition of the rst derivative to nd the instantaneous rate of change of
f (t) = t2 3t
at t = 4.
Solution
The derivative of f (t) is
(t + h)2 3(t + h) t2 3t
f (t + h) f (t)
f t() := lim
= lim
h0
h0
h
h
t2 + 2th + h2 3t 3h t2 3t
=
Question 1
graph of the curve y = x2 + 2x + 15. Hence, the x-axis and the lines x = 4 and x = 3.
Solution
Using y = x2 + 2x + 15 = (x + 3)(x 5)
we obtain the graph shown.
The area of the bounded region (shaded region) is A1 + A2
where
3
A1 =
f (x) dx
4
1
Question 1
A particle is conned to move along a straight line. Its position, s metres after t seconds from a xed point, is
given by
s(t) = 2t3 15t2 + 24t + 2.
Determine when the particle is momentarily at rest, and nd the acceleration at each of these tim
Question 1
If yx = 3x2 + 7x + 20 and y = 11 when x = 3 nd y in terms of x.
Solution
On integrating yx we obtain
y = x3 + 7 x2 + 20x + c.
2
Using the initial condition, y(3) = 11, we can solve for constant, c.
11 = (3)3 + 7 (3)2 + 20(3) + c c = 107.5
2
Hen
Question 1
State the domain and range of the function
f (x) = 2 + ln(x + 3).
Hence sketch the graph of y = f (x).
Solution
Domain of f (x)
Division by zero
never occurs.
Square root of negative number
never occurs.
Log of non-positive number
occurs
Question 1
Sketch the graph of the curve y = x2 + 2x + 15. Hence, determine the area of the nite region bounded by
the curve y = x2 + 2x + 15, the x-axis and the lines x = 4 and x = 3.
Solution
Using y = x2 + 2x + 15 = (x + 3)(x 5)
we obtain the graph sho