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Sample Final Exam Questions Precalculus M191 Prof.
L. A. Month
DO NOT WRITE ON THE EXAM
Write all your answers on the paper provided. SHOW ALL WORK.
Don’t forget to write your name on all your papers.
You may use your graphing calculator and your notes.
(3) 1. Solve |x–2|<5. Write your answer in interval form
(5) 2. Graph |x–2|<3x. Solve for x. . Label all curves and relevant intersections.
(5) 3. x2y–x3+3=0 Find: x and y intercepts, domain, range, asymptotes.
(5) 4. Classify as odd, even or neither:
x
x + x7
1 − x2
2 + x2
x2 + x
ex + e− x
2
ex − e− x
2
ex
cos 2x
sin ( −x)
tan −1 x
sec −1 x
(4) 5. In how many places do the graphs of y=x2 and y=2x intersect? Find the intersections.
(5) 6. Which of the following functions have no turning points?
ln x, ex, x, x2, cos x, sin x, 3–x, x(x+1), log2x, (sin x)2
(10) 7. Solve
x +1 x − 3
≥
x + 4 x −1
analytically. Sketch the graphs. Shade the desired region.
(10) 8. Sketch
(x–r1)4
(x–r1)2(x-r2)2
(x–r1)3(x-r2)2
(x–r1)(x-r2)2(x–r3)
(5) 9. f(x)=2+cos(2x)
Find: f–1(x), domain and range of f–1(x).
(5) 10. Start with f(x)
To obtain g(x): translate to the left 2 units; reflect about the x axis; translate up 3 units;
reflect about y=x.
If (a,b) starts on f(x) where does it end up on g?
If f(x)=x2, find g(x). Sketch g(x).
(5) 11. Which of the following compositions are linear? Which are quadratic?
f(x)=x2 g(x)=x h(x)=ex k(x)=ln x
f°g g°f h°g k°h f°k
(5) 12. Solve log2(3x+2)>log4(25x).
DVC Precalculus Sample questions for final
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(5) 13. Solve
3x = e x
2
A
1 + ae − kt
What is the name of this curve? What is the growth rate for x? What is x0? What is x
as t→+∞?
Find a, A, and k if x(0)=4, x→+5 as t→+∞, and x=2 at t=1.
(5) 14. Sketch
x=
(5) 15. For quadrant II:
Give the signs of the 6 basic trig functions.
For each, state whether they are increasing or decreasing.
Which of the inverse trig functions have ranges in quadrant II?
What is the interval of all angles (in radians) in quadrant II.
(5) 16. Evaluate:
(5) 17. Verify
tan(–π/6)
sec(–2π/3)
cos(–3π/4)
csc(5π)
cot(6π)
1 + cos θ
cos θ
=
. State the domain.
2
−1 + sec θ
tan θ
3
–∞<x<∞
3
Sketch the roots on the unit circle.
(5) 18. Solve
tan x =–
(2) 19. Which is bigger, 1 radian or 1 degree? Why?
(4) 20. Find all solutions of
a) θ=tan–1(–2)
b) tanθ=–2
(5) 21. Which functions are the same as sin(2x)?
–sin(2x) –sin(–2x) cos(2x+π/2) cos2(x–π/2)
2sin x cos 2x cos(2x–π/2) cos(π/2–x)
cos2(2x)–1
(5) 22. Find sin(cos–1(1/3)). Leave your answer in rational form.
Solve cos–1x=sin–1x
(5) 23. Find sin(π/12). Leave your answer in rational form.
(2) 24. Calculate the determinant of
 1 30 
A= 2
.
 50 600
(4) 25. Which of the following are equivalent to
2 1 4
23 2 9
5 3 7
2 3 4
3 6 7 ?
5 9 9
3 6 7
5 9 9
2
3
5 9
2 3 4
−5 − 9 − 9
9
DVC Precalculus Sample questions for final
L.A. Month
3
6
−4
−7
2 3 4
5 9 11
(5) 26. A =(2,3) in rectangular coordinates
B =5 i$ +6 $j
Evaluate A + B and the angle of B. Verify the triangle inequality for A + B .
(5) 27. Find the tangent to the circle x2+y2=169 at (5,12). Graph it. Write your answer in
rectangular and polar forms.
(6) 28. Find the slopes of the lines
(y-3)=5(x-2)
θ=π/6
y=3x+2
r cos(θ-π/4)=5
2x+3y+5=0
r(sinθ–3cosθ)=2
(5) 29. Given the graph of r=2+3cosθ:
Draw arrows to indicate how r changes
as θ increases from 0 to 2π.
Evaluate r at θ=0, π, and 3π/2.
For 0≤θ<2π, when is r increasing?
Sketch r=2–3cosθ.
8z+1 = 32 2
(5) 30. Solve
(5) 31. Show that the complex numbers z=x+iy=(x,y) form a vector space with vector addition
defined as z1+z2=(x1+x2,y1+y2) and scalar multiplication defined as αz=(αx,αy),
where α is a real number.
(5) 32. If z=–6–17i
a. Graph z.
b. Write z in polar form.
c. What are the argument and modulus of z.
d. What is the real part? The imaginary part?
e. What is the argument of z10 ?
f. What is the modulus of z3 ? In what quadrant is z3 ?
(5) 33. Show that the complex conjugate of z=(r,θ) is (r,–θ).
(5) 34. Sketch z1=–2+3i and z2=4+5i.
Find z1+z2 in polar form.
Find z1×z2. Sketch z1×z2.
Find z1+z2 in rectangular form.
Find the argument of z1÷z2.
(5) 35. Find
4
−16 . Sketch the results.
(5) 36. Given f(x)=2x3+3x2+2x+8.
Find the remainder when f(x) is divided by (x+2).
Find the remainder when f(x) is divided by (x+1).
DVC Precalculus Sample questions for final
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(5) 37. Find a polynomial of degree 3 which has a coefficient of the x3 term equal to 5 and three
roots 1, 2, and 3.
(5) 38. Factor 2x3+3x2+2x+8.
(5) 39. Write the partial fraction decomposition for
x3
x5
x4 + 4x2 + 4
x4 + 4x2 + 4
You need not solve for the constants.
(5) 40. Given the polynomial equation 10x4+5x3–3x2+6x+1=0.
How many positive roots does it have? How many negative roots?
How many rational? How many irrational? How many complex?
BONUS
(5) 1. Find (2+3i)1/3 in rectangular form.
(5) 2. Find all the roots of the polynomial given in problem 40. If you need to estimate any
roots, estimate them to the nearest hundredth using the method described in class.
2( x − 3)
y=f(x)=
( x − 5)( x − 2)
2
(5) 3. Graph the function
Specify the domain, range, intercepts, and asymptotes.
Indicate and find the turning points.
Sketch the behavior as x→±∞ and near the asymptotes and zeroes.
(5) 4. z1=2+3i and z2=(r,θ)=(3,30°)
Show that
z1 + z2 = z1 + z2
z1 × z2 = z1 × z2
3
3
z2 = z2
DVC Precalculus Sample questions for final
L.A. Month