Polynomials
Determine the x-intercepts of the function:
y = 3x^2 - 5x - 12
x = 3
x = -4/3
State the domain and range of f(x)=sqrt(x+3)
domain = [-3, infinity)
range = [0, infinity)
Differentiatef(x) = 3x^2 - 5x + 4
f'(x)=6x-5
Find the exact value of sin(-13pi/6)
-1/2
Evaluate 5 x 32 x 34
3645
Find the remainder when 2x^3−5x^2+4x−7 is divided by x−2.
remainder = -3
Write down the equation of the hyperbola with asymptotes at x=0 and y=0, passing through the point (2,3).
y = 6/x
The position of a particle is given by s(t) = t^3 - 6t^2 + 9t. Find the velocity at t=2.
s'(2)=-3
Solve cos(2x)=0 for -pi < x < 2pi
-3pi/4, -pi/4, pi/4, 3pi/4, 7pi/4
Sketch f(x)=2x−1 for -2<x<2
Correct sketch -
asymptote at y=-1
end points (-2, -3/4) and (2, 3)
Factorise completely:
x^3 - 4x^2 - 7x + 10
(x+2)(x-1)(x-5)
The function f(x)= sqrt(x^2 - 4x + 3) is given.
(a) Determine the maximal domain of f(x).
(b) Find the x-intercepts, if any.
(a) (-infinity, 1] u [3, infinity)
(b) x=1, x=3
Find the equation of the tangent line to y = x^2 - 4x + 5 at x=3.
y=2x-4
The function f(x)=2cosx−1 is given for 0≤x≤2π.
(a) Determine the amplitude, period, and vertical shift.
(b) Find the x-values of the maximum and minimum points in the interval.
(c) Sketch the graph over 0≤x≤2π.
(a) amp = 2
shift = 1 up
period = 2pi
(b) max = 0, 2pi
min = pi
Solve 4x- 5 x 2x = - 4
x=0, x=2
The polynomial P(x) = x^4 - 3x^3 - 4x^2 + 12x has x=2 as a root.
(a) Use this fact to factorise P(x).
(b) Hence, find all the roots of P(x).
(a) x(x-2)(x+2)(x-3)
(b) x=0, x=2, x=-2, x=3
The hyperbola f(x)=(3x+2)/(x−1)
(a) Determine the vertical and horizontal asymptotes.
(b) Find the x- and y-intercepts of the graph.
(c) State the domain and range.
(a) x=1, y=3
(b) x=-2/3, y=-2
(c) Domain = R\{1}
Range = R\{3}
The function f(x) = x^3 - 6x^2 + 9x + 1
(a) Find the x coordinates of the stationary points.
(b) Determine the intervals of increasing/decreasing.
(c) Identify local maxima and minima.
(a) x=1, 3
(b) Increase = (-infinity,1) u (3, infinity)
Decrease = (1,3)
(c) (1, 5) and (3, 1)
A sine function is given by y=3sin(2x−pi/2)
(a) Determine the amplitude, period, and phase shift.
(b) State the coordinates of the first positive peak.
(a) amp = 3
period = pi
pi/2 to the right
(b) (pi/2, 3)
A radioactive substance decays according to M(t)=200 x (1/2)t where t is measured in hours.
(a) Find the mass after 5 hours.
(b) Determine how long it takes for the mass to reduce to 25 g. (Exact form acceptable without logarithms)
(a) 6.25 or 25/4
(b) t=3 hours
A cubic polynomial P(x) with leading coefficient 1 has roots at x=−1, x = 2, and x = k. It also satisfies P(1)=12.
Find the value of k and the equation of the polynomial in standard form.
k=7, P(x)=x^3−8x^2+5x+14
f(x) is a piecewise function with the following branches:
f1(x)=x+2, x=<0
f2(x)=sqrt(x)
(a) Sketch the graph
(b) Determine whether f(x) has an inverse and explain.
Not one-to-one over the whole domain, so no global inverse.
A rectangular garden is to be built against a straight wall. The gardener wants to enclose a total area of 200 m² using fencing on three sides only (the wall forms the fourth side). Determine the value of x that minimises the fencing required and find the corresponding length and the minimum total fencing needed.
length along wall = 20m
Minimum total fencing =40 m
Solve the trigonometric equation 2cos^2(x) - cos x - 1 = 0 for -pi<x<2pi
x = 0, 2pi, -2pi/3, 2pi/3, 4pi/3
Solve the equation 2x+1+2x=12
x = 2