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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