Familiar Functions
Trigonometry
Derivatives
Probability
Calculus Applications
100
Consider the rational function below. State the equation of the vertical and horizontal asymptotes.
f(x)=(4x-1)/(2x+6)
x=-3 and y=2
100
Given that
cos(x)=-2/3
and
pi<x<(3pi)/2
find the exact value of sin(2x)
(4sqrt5)/9
100
Derive:
3x^2-1/x^3+sqrtx
6x+3x^-4+1/2x^(-1/2)
100
It is given that
P(A)=0.6, P(B)=0.7, and P(AUB)=0.9
Determine and justify whether events A and B are independent.
No, they are not independent
100
Given the graph of f'(x), identify the exact interval(s) of decrease.
x<-3
200
A quadratic function, f, has a vertex at (2,9) and passes through the point (5,0). Find the values of p and q if the function is expressed as
f(x)=(x-p)(x-q)
p=-1, q=5
200
A ship leaves port P and sails on a bearing of 060 degrees for 15 km to reach point A. It then changes course and sails to point B. The distance AB is 10km.
If the bearing of B from P is 080 degrees, calculate the distance BP.
22.7km or 5.51 km
200
Find
dy/dx of y=cos^2x
-2cosxsinx = - sin2x
200
In a group of students, 60% play football (F) and 45% play basketball (B). It is known that 20% of the students play neither sport. A student is selected at random. Given that the student plays football, find the probability that they also play basketball.
5/12=0.417
200
Find the equation of the normal to the curve f(x) = ln(x) at the point where x = e.
y-1=-e(x-e) or y=-ex+e^2+1
300
Find the x-coordinates of the two points of intersection of the functions
f(x)=(2x-1)/(x+3)
and f^-1(x)
x={-1,1}
300
A right pyramid has a square base ABCD with side length 8cm. The vertex V is directly above the center of the base, O. The angle between a slanted EDGE and the base is 60 degrees. Calculate the exact height, VO, of the pyramid
4sqrt6
If you used the slant side, 4sqrt3
300
Find
f'(pi/4)
given f(x)=tanx
f'(x)=1/cos^2x f'(pi/4)=2
300
A discrete random variable X has the following probability distribution:
P(X=x)=k(x^2+1) when x={2,3,4}
Find the expected value of E(X).
27/8=3.375
300
Given the function of f(x). Determine the coordinates and nature of any stationary points.
f(x)=x^4-4x^3
Local Minimum at (3, -27) and Stationary Point of Inflection at (0, 0)
400
The graph of f(x)=lnx is translated down 1, left 2, and then reflected over the x-axis to produce the graph of g(x). Show that g(x) can be written in the form
g(x)=ln(e/(x-2))
-(ln(x-2)-1)
1-ln(x-2)
lne-ln(x-2)
ln(e/(x-2))
400
The function
f(x)=asin(b(x-c))+d
has a maximum point at
(pi/12,7)
and the next minimum point at
((5pi)/12,-1)
Find the values of a, b, c, and d, where a, b >0 and
0<c<pi
a=4, b=3, c=(7pi)/12, d=3
400
Differentiate
(e^(2x)lnx)^3
3(e^(2x)lnx)^2(e^(2x)/x+2e^(2x)lnx)
400
In a large shipment of lightbulbs, 5% are known to be defective. A random sample of 20 bulbs is selected. Find the probability that more than three bulbs are defective. Give your answer to 3 significant figures.
0.0210
400
Find the value of k, given that the line given below is normal to the curve f(x) at
x=pi
y=1/2x+5
f(x)=ksin(x)
500
A function h is defined below for x>k+1. The graph of y=h(x) has a horizontal asymptote at y=3 and a vertical asymptote at
x=e^2+5
Determine the exact value of the x-intercept of the graph h
h(x)=(aln(x-k)+12)/(ln(x-k)-2)
k=5, a=3
(e^-4+5,0)
500
Find all exact solutions for x in the interval
-pi<x<2pi
for the equation
4sinxcosx=2cosx-2sinx+1
x={(-2pi)/3,pi/6,(2pi)/3,(5pi)/6,(4pi)/3}
500
Given the function below, find y' in terms of f and g
y=f(g(2x))
2f^'(g(2x))(g^'(2x))
500
The weights of bags of flour are normally distributed with mean, a, and standard deviation, b. It is known that 10% of the bags weigh less than 490g and 5% of the bags weigh more than 515g. Find the value of a and the value of b.
a= 501 and b=8.54
500
Consider the function f(x) below, where k is a constant. Given that there is a point of inflection at x=2, find the coordinates of any stationary points.
f(x)=x^3+kx^2+12x-5
Answer: (2,3)
