Advanced Functions Exam Review
Part A: Short Answer
1. List the asymptotes of the following:
2 3x 2 3x x2 5x
a) y b) y 2 c) y
6 x x 5 x 14 x 1
2 3x
2. Solve: 5
6 x
x3 5x
3. Describe the function y 7 as even, odd or neither
6 x 4 x3
4. An odd function has 3 vertical asymptotes, one is x = 3, what are the other two?
5. T or F? An even function that is continuous at x = 0 has a local max or min there.
6. T or F? A reciprocal function has no roots.
7. Convert to radians. a) 225o exact b) 164o 3 decimal
7
8. Convert to degrees a) b) 2.34 (1 decimal)
12
9. Determine the angle x [0, 360] and [0, 2л]
a) sin x = -0.5 b) cot x = 3 c) sec x = 2.5 (1 decimal)
1
d) cos = e) csc = 2 f) cot = 2.5 (1 decimal)
2
10. Solve for the angle x [0, 360o] and [0, 2л]
2
a) csc x = 2 b) 3sec - 4 = 0
2
c) sin2x = 0.5
11. Express as a simple trig function of the angle x.
3
a) sin x b) sec( x) c) tan x
2 2
12. Give the period, amplitude, phase shift and axis of y = 5sin(3x – л) – 7
13. Simplify.
a) sin 13cos25 sin 25 cos13 b) 2 sin 50 cos50 c) cos2 x 1
d) cos(a 2b)cos(a b) sin(a 2b)sin(a b)
14. Give an exact value for the csc(15o).
1000t 2
15. The population of trout in a river is given by N (t ) 1000 2 , t 0.
t 100
a) What size will the trout population be after a long time?
b) How many trout were in the river to begin with?
c) How fast is the trout population growing at three years?
d) What is the average population growth for the first three years?
16. The probability, P of hitting a target x feet away is graphed below.
a) What is the average rate of change of P as a player moves from 10 ft to 90 ft away?
b) How fast is the probability changing when the participant is 50 ft away?
110
100
90
80
70
60
50
40
30
20
10
-20 20 40 60 80 100 120 140 160 180
17. By calculating finite differences, determine which type of function best models this
relationship.
input –2 –1 0 1 2 3 4
output 1 0 1 16 81 256 625
18. f(x) = –x2 + 16 and g(x) = x – 4. Calculate and .
19. What is the remainder when x4 – 5x3 + 5x2 + 4x – 4 is divided by x – 2?
20. a) Evaluate: Solve for x:
b)
21. Express, in terms of x, the slope of any secant through (1, 1) on the graph of .
22. The position of a particle is given by the function . Estimate the
particle’s instantaneous rate of change of distance at 1 s.
23. Let f = {(0, 2), (1, 4), (2, 6), (3, 8), (4, 10)} and g(x) = 3x – 1. State the domain of f(g(x)).
24. Let h(x) = (x2 + 3)3. Determine two functions f and g such that .
Part B: Long Answer
1. Solve ( [0, 2л])
a) 2x4 + 4x + 4 = x3 + 9x2 b) x4 + x3 + 3x + 7 > 5x2 + 7
x2 2x 5 4x 1 2x 1
c) 2 3 d) 2
2 x 5x 3 4 x 9 4 x 4 x 2 9 x 9
2
x 4 x2
e) 8.72(0.93)x + 3 + 17 = 22 f) log12(x – 3) + log12(x + 1) = 1
f) cos(2) + 5 = 4sin2() + cos() + 2 g) 2x – x2 = 0
2. Show that the line y = -10x + 20 is tangent to the curve y = x4 – 4x3 – 5x2 + 26x – 16.
4. Stan invests $1000 at 3% compounded semi-annually and $1500 at 1.8% compounded
annually. When will the two investments be equal?
5. Determine whether the following are equations or identities. If they are equations
solve them, otherwise prove the identity. x [0, 360o]
a) 4cos2x = 3 – 2sin2x b) sin4x + cos4x = sin2x(csc2x – 2cos2x)
1 sin 2 x cos 2 x
c) cos2 x 1 sin 2 x d) (cot x)(csc x)(tan x)(cos x) = cos 2x + 2sin2x
12 4
sin x , 0 x cos x , x0
6. If 13 2 and 5 2 , determine an exact value
for sin[2(x - y)].
7. Describe the relationship for functions with degree 1 through 4:
a) between the degree and the number of turning points
b) between the sign of the leading coefficient and the end behaviour as x
Corners are cut from a 30 cm by 20 cm piece of cardboard. The volume is given in terms of
the size of the square cut out, by V(x) = x(30 – 2x)(20 – 2x) where the height is x.
a) Calculate the volume when the height is 2 cm.
b) Calculate the dimensions of a box with volume 1000 cm3.
9. A mass on the end of a spring is pulled so that its distance from the rest position is
initially 3 cm. After being released the mass oscillates while the spring contracts and
expands. The motion of the mass is sinusoidal with a period of 3 seconds.
a) Give the theoretical equation for the mass of its distance from rest in terms of the
time t, in seconds, assuming no energy is lost with each cycle.
b) If the spring looses 5% of its energy with each cycle, give the equation for this
motion.
10. The population of an Ontario town grew from 1250 to 10 000 in 5 years due to the
establishment of a large industry in the area.
a) Determine the doubling period of this population.
b) Derive an expression for the population of this town after t years.
c) Determine the population after 3 years.
11. The 1970 earthquake in Peru registered 7.7 on the Richter scale. How much more intense was
this earthquake compared to the tremor in Ontario in 2002 that was measured to be 5.2 on the
Richter scale?
12. Determine a logarithmic equation that models the following data and expresses y in terms of
x and calculate the missing value.
x 1 10 100 1000 10 000 100 000
y 0 5 10 15 20
13. A round balloon is being blown up in such a way that after t seconds its radius has grown to
2t cm. Reminder: The volume of a sphere of radius r is and the surface area is .
a) Find the surface area of the balloon as a function of t.
b) What is the average rate of change of the surface area with respect to the time over the first 5
seconds?
c) How fast is that changing when the radius is 10 cm?
14. The frets on a guitar are placed so that they make the correct vibrating string length
for the note of music. We are interested in how the vibrating string length changes for
each fret position. Below is the length from the bridge to each fret position.
Fret Number 0 1 2 3 4 5 6 7 8 9 10 11 12
Length (mm) 660 623 588 555 524 494 467 440 416 392 370 350 330
Using one of the methods form class determine the relationship between the fret
number and the length.
15. Sketch a function f(x) that has all of the following characteristics;
lim f ( x) lim f ( x) lim f ( x) lim f ( x)
x1 x1 x4 x4
2
lim f ( x) 2 lim f ( x) 2 x int er 7 y int er
x x 3
16. I have $100 dollars to invest and I want to know how to allocate it between two
possible entrepreneurs, Alpha and Beta, in order to maximize my total annual return.
x(200 x)
Alpha: If I give x dollars to Alpha, my annual return is A( x) .
1000
Beta: If I give the remaining (100–x) dollars to Beta, my return is B(x) = r(100–x)
dollars per year. That is, Beta simply pays me annual interest at rate r.
(a) Take r=12% (that is, r=0.12). Using the given formula for A(x), find the allocation
which maximizes the sum of my returns through Alpha and Beta. Illustrate your
solution on a copy of the graph.
(b) In case the optimal allocation is split, find a formula for the optimal allocation x in
terms of the interest rate r. What interest rates r would compel me to give everything
to B?
17. After you eat something that contains sugar, the pH or acid level in your mouth
20.4m
changes. This can be modeled by the function L(m) 2 6.5 , where L is the
m 36
pH level and m is the number of minutes that have elapsed since eating. Find the
average rate of change from 1.5 minutes to 3 minutes, and find the instantaneous rate
of change at 3 minutes.
Part C: Graphing
1. List the key properties of each graph. Use these to create a sketch without technology.
Indicate at least two key points on each curve.
2x 4 1
y y 2
3 2
a) y = -2x – 6x + 8 b) x 5 c) x 4
d) y = 3tan(2x – ) + 4 e) y = 2sec(0.5x) f) y = 3-x + 5
g) y = 2log6(x – 5) h) y = x + cot(x)
Part D: Reverse Graphing
1. For each sketch below;
a) List the key properties of the graph.
b) What type of relationship is being shown?
c) Use a guess and check method to determine the equation of the curve.