Math 1080 Additional Practice Problems Midterm 2
Find the rational zeros of each polynomial.
1. p(x) = 4x3 − 3x2 + 4x − 3
2. 5x6 − 26x5 + 30x4 − 70x3 + 33x2 + 136x − 28
Find the real zeros of each polynomial.
1. x4 − 10x2 + 9
2. r(x) = x3 − 8x2 + 29x − 52
Find the complex zeros of each polynomial.
1. p(x) = 5x6 − 26x5 + 30x4 − 70x3 + 33x2 + 136x − 28,
√ given 2i is a zero of the polynomial.
2. f (x) = x5 + x4 − 9x3 + 7x2 − 52x + 12, given −2 + 5 is a zero of the polynomial.
3. g(x) = x3 − 8x2 + 29x − 52
4. q(x) = 4x3 − 3x2 + 4x − 3
5. r(x) = 4x5 − 4x4 − 13x3 + 13x2 + 3x − 3
1. Find the coefficient for the a4 b4 term in (a + 2b)8 .
Expand each Binomial, using the Binomial Theorem.
2. (x − 1)10
1
3. (x 2 + y 3 )4
Find the domain of each function.
2
+2
1. f (x) = xx−1
2
x −3
g(x) = √
x+2
f (t) = et−2 + 4
g(x) = log(x) − 2
1
r(x) = (x2 − x − 6) 4
f (s) = s4 − 10s2 + 9
2
−7
h(x) = 4x
5x
q 2 +1
4x
8. p(x) = x2 +2x−6
9. f (x) = ln(x + 6)
1
10. g(z) = (1 − 4z) 3
2.
3.
4.
5.
6.
7.
Find the x-intercepts and y-intercept of the each function.
2
−7
1. f (x) = 4x
5x2 +1
3x+1
2. f (x) = 2e
−5
3. f (t) = 3t3 − t2 + 3t − 1
4. f (x) = √
log7 (5 − 4x) − log7 (x2 )
5. g(z) = 1 − 4z
6. f (x) = −2x2 + 24x − 1
Find the asymptotes of each function.
1. h(t) = log(t) − 2
2. f (x) = ex−2 + 4
1
3. g(x) =
4. g(z) =
5. f (x) =
6. h(x) =
x2 +2
x−1
4z 2 −7
5z 2 +1
5x
x2 +1
2x3 +x2 −15x−18
x2 −x−6
Sketch the graph of each of the following functions. (Check your work with a graphing calculator.)
1. f (x) = ex−2 + 4
2. f (x) = log(x) − 2
2
−7
3. f (x) = 4x
5x2 +1
4. f (x) = −2x2 + 24x − 1
5. f (x) = ln(x + 6)
Solve
√ each equation.
1. x + 2 + 3 = 7
2. log5 (x − 10) = 2
3. log (3x + 3) − log (x + 5) = log(x)
4
3
− 2x−1
= 25
4. 3x+1
√
√ 6x −x−1
5. 2x = 1 − x + 1
6. 3e2x+1 − 1 = 11
7. log
√ 4 2 + log4 8 = x
8. 3x − 5 = x − 1
8
1
9. x+6
− x−4
= 13
10. 52+x = e3x+2
11. 32x + 3x = 20
12. log (x2 ) + 15 = [log(x)]2
13. log (x) + log5 x = 5
14. logx (x + 1) = 2
15. ln (log (x − 1)) = 0
x
−x
16. e −e
= 50
2
1. You make a one time deposit of $250 into a savings account which yields an annual interest rate of
4.25% compounded monthly. When will your account contain $5,000?
2. You would like to have $200,000 in an account in 25 years. If you can get an account with an annual
interest rate of 5.12% compounded quarterly, how much must you deposit now so that the future value of
the account will be $200,000 in 25 years?
3. You invest $2000 in and account with an annual interest rate of 8% compounded every two months. What
is the future value of your account in 10 years?
4. The cost of producing x units is C(x) = 5x2 − 10x + 15600. What is the minimum cost?
5. An indoor physical fitness room consists of a rectangular region with a semicircle on each end. The
perimeter of the room is to be a 200 meter running track. Write the area of the room as a function of only
one variable.
6. A healing law for skin wounds states that A = A0 e−0.1t , where A is the number of square centimeters of
unhealed skin after t days when the original area of the wound was A0 . How many days does it take for half
the wound to heal?
7. The half-life of a radioactive material is 25 years. How long till only 10% of the initial amount of material
2
remains?
(x)
Find f (x+h)−f
and simplify.
h
√
1. f (x) = x + 1
2. f (x) = x2 − 3x
3