Review questions for Math 111 final. Please SHOW your WORK to receive full credit Final Test is based on 150 points

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1 Please SHOW your WORK to receive full credit Final Test is based on 150 points 1. True or False questions (17 pts) a. Common Logarithmic functions cross the y axis at (0,1) b. A square matrix has as many rows as columns c. f(x)= log b (x) is an increasing function (b= 2 is the base value) d. Line L1 and L2 are parallel if their slopes when multiplied equal to -1 e. The inverse of Ln (x) is e x f. A zero Matrix has all its elements equal zero g. An identity Matrix must be a square Matrix h. The range of Log x function is (-, + ) i. The domain of e x is (0, ) j. When two matrices A 1x4 is multiplied by matrix B 4x10, the resultant is matrix C with that has 5 rows and 14 columns. k. Synthetic division cannot be applied to divide any two polynomials. l. The leading term of a polynomial is the one that dominates the polynomial behavior when x goes to infinity m. A polynomial with real coefficients can only have real zeros. n. The multiplication of a complex number with its conjugate pair yields a real number o. The value of a rational function at its vertical asymptote is undefined p. Inverse functions have symmetry across the y = x axis q. A one to one function will always have an inverse function. 2. A model rocket is launched upward with an initial velocity of 22 feet per second. The height, in feet, of the rocket t seconds after the launch is given by the equation h(t) = -16t t. How many seconds after the launch will the rocket be 350 feet above ground? Round your answer to the nearest tenth of a second. (6pts) 1

2 3. Solve the following inequality (5 pts) 4x 3 > 5 4. Find the maximum or minimum value of each of the quadratic function. (show your work) (6pts) a. F(x) = x 2 + 4x b. f(x) = -2x 2 + 4x Solve the following equation by finding all the zeros of the f(x) (7pts) f(x) = 2x 4 + 3x 2-2 2

3 6. Determine the domain of each of the following functions (6pts) a. G(t) = t 2 t 2 3t 10 b. H(x) = 1 4+x 2 7. Find the equation of the line L1 that passes by point P (-6, -2) and is also parallel to L2 where L2 is represented by equation 2x-3y = 7. (6pts) 3

4 8. Give the following function f(x) (6pts) Where x 4 f(x) = x + 1 x 4 a. Find the inverse function f -1 (x) of the following equation: b. What are the vertical and horizontal asymptotes for the above function 9. Find the quotient and remainder using the following two methods when dividing P(x) by (x-3) where P(x) is given by: (12pts) a. Long Division P(x) = 2x 4 + 3x 3 + 2x - 2 b. Synthetic Division 4

5 c. Using the remainder theorem, what would be remainder of P(x) when divided by (x-3)? d. How many possible positive and negative zeros can the above equation have? Explain your answers to why? 10. A polynomial P(x) is of 4 th degree and has real coefficients with two of its roots as 3i, and 2-i. (6pts) a. What would be the values of the other two roots? b. Write down P(x) in polynomial form given the constant factor is equal to one. 5

6 11. Write the following equation in exponential form. (4pts) a) 3+a = log 10 (5x+2) b) Log e x = 4b Write each equation in it logarithmic form (4pts) a) e 5 = 3x+1 b) y + 3 = e x 13. Find the domain of the following functions (5 pts) a) f(x)= log 5 (x 2 +3) + e 2x b) g(x)= 3 x 20 2(x 2 1)(x+5) 6

7 14. Use the properties of logarithms to expand the following logarithmic expressions. Assuming all variables expressions represent real numbers (8 pts) a) Log( (x3 z 2y ) b) Ln ( e3x 2xy 2) 15. Use the properties of logarithms to rewrite each expression as a single logarithm with a coefficient of 1. Simplify your answer where applicable (8pts) a. -2log (x+1) + log y +3 log z + log (x 2-10) b. Ln (x - 4) - ln(x 2-4) + Ln(x + y) 7

8 16. Solve the following logarithmic equation to find the value of x (6pts) Log (14x) log (x+2) = Triple Your Money (6pts) Find the time required for money invested at an annual rate of 4% to triple in value if the investment is compounded continuously. The balance after t years is given by: A= Pe rt. Where P is the principal, t is the number of years, and r is the annual interest rate. 18. The number A of cars sold annually by an automobile dealership can be closely modeled by the logistic function (8pts) 8

9 1650 A(t) = 1+2.4e 0.055t Where t is the time in years since the dealership was founded. a.. How many cars will the dealership sell during its first and second year of operations? b. According to the model, what will be the dealership annual sales in the long-term future? 19. Solve the following system of equations using an augmented Matrix and Gaussian elimination. (Hint: You need to put the resulting matrix in Echelon format) (8pts) 2x + 4y + 6z = -2 5x + 7y - 13z = 2 4x + 6y + 10z = 3 9

10 20. Given the two Matrices shown below (8 pts) A= , and B = Find the followings: a. A + B = b. 2A - 3B = c. A x B = d. A x I 2, where I 2 is the identity Matrix. 21. Find the values of: a, b, c, d, e x, y, and z given that both matrices A and B (shown below) are equal: (8 pts) 2x z 10 h 10 3 [ 20 a 24] = [ c ] 10 0 b d y e 10

11 22. Bonus Questions (Total of 8 points) 1. (4 pts) a) Two equations are labeled identical if they have infinite solutions b) Logarithmic functions cannot take negative values. c) The multiplication of a matrix by a scalar (number) does not changes the size of the matrix. d) Inverse functions must pass the vertical line test 2. (4 pts) The distance sound travels vary directly as the time it travels. If sound travels 1340 meters in 4 seconds, find the distance sound will travel in 5 seconds. 11

Mock Final Exam Name. Solve and check the linear equation. 1) (-8x + 8) + 1 = -7(x + 3) A) {- 30} B) {- 6} C) {30} D) {- 28}

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