2018 MIDTERM EXAM REVIEW

Transcription

1 Name: Hour: 2018 MIDTERM EXAM REVIEW PRE-CALCULUS Please keep in mind that this exam is worth 20% of your overall grade for this SEMESTER and your semester grade is averaged into your overall GPA. Schedule of Assignments Chapter/Unit Due Date 1 - Solving Monday, January 8 th 2 - Functions Tuesday, January 9 th 3 Graphing Wednesday, January 10 th 4 Exponential/Log Functions Thursday, January 11 th 5 & 6 - Trig Friday, January 12 th 7th Hour Exam: THURSDAY, JANUARY 18 th (9:45-11:30)

2 UNIT 1 Solving Equations Solve the equation. (Non-Graphing Calculator) 1. x 3 + 4x 2 x 4 = 0 2. x 2 7x + 4 = 0 3. x 3 + 3x 2 4 = x 3 25x 2 15x = x 2 + 8x 5 = 0 6. x 3 + 4x 2 + 3x 2 = (x + 1) 2 + 5(x + 1) 3 = 0 8. x 2 + 2x + 2 = 0

3 9. x 4 6x 3 + 4x x + 4 = (x 2 + 5x) = x x 3 x 2 + 6x 2 = x 4 13x = x 4 4x x 2 24x + 24 = Suppose a bullet was fired into the air with an initial velocity of 848 ft/sec. Its height h after t seconds is given by h(t) = 16t t. (CALCULATOR) a) When will the bullet reach 10,000 ft? b) When will the bullet reach 12,000 ft? c) When will the bullet hit the ground?

4 15. A ball is thrown across a baseball field. Its path is modeled by the equation h = 0.005x 2 + x + 6, where x is the distance the ball traveled horizontally and h is the height of the ball. Both x and h are measured in feet. (CALCULATOR) a) What is the maximum height attained by the ball? b) How many feet horizontally does the ball travel when it reaches its maximum? 16. A farmer has 600 ft of fencing to fence in a pen for his goats. He plans to create a rectangular pen along the side of his barn and will not need to install the fence along the barn. What is the maximum area he can enclose? (CALCULATOR) 17. Find a 3 rd -degree polynomial with zeros 2 and 5 and with a leading coefficient of 4. (CALCULATOR) 18. Find a polynomial that has zeros 6 and 1 i. (CALCULATOR)

5 Simplify the expression. (CALCULATOR) 19. 2x3 x 2 6x 2x 2 7x x2 +5x+4 x 2 3x 4 4 x x x2 7x 4 x 3 5x 2 x2 x 12 x 2 +3x 22. x + 5 x+5 x x 3 x 2 +x 2 x 2 5x x+7 2 x+7 1 x 26. xy 2 yx 2 x y 2 1 h h Solve the equation. (CALCULATOR) = 4 x x x x = x 2 x x x = x 1 x + 4 x 3 x 3

6 UNIT 2 Functions Given f(x) = x 2 + x 12 and g(x) = x + 4, evaluate the following. (CALCULATOR) 1. f( 2) 2. f( x) 3. g(x + 2) 4. f(2) + g(x) 5. 5f(x) 6. f(x + h) 7. g(x) f(x) 8. f(x)g(x) 9. ( g f ) (x) 10. (f g)(0) 11. (g f)(x) 12. (g g)(x) Evaluate the difference quotient for the given function. (CALCULATOR) 13. f(x) = 2 x f(x) = 3 x 15. Express x 3 in interval notation. 16. Express x 6 in interval notation. 17. Express (, 5) using inequalities.

7 The graph of f is given. Use the graph to identify the domain and range of f in interval notation. State the interval(s) on which f is increasing, decreasing, or constant. Then tell whether f is one-to-one. (CALCULATOR) 18. Domain of f: Range of f: Increasing: Decreasing: Constant: One-to-One: 19. Domain of f: Range of f: Increasing: Decreasing: Constant: One-to-One: Find the domain of the function algebraically. Write your answer in interval notation. (Non- Graphing Calculator) 20. f(x) = 5 x 21. f(x) = 1 x f(x) = x x f(x) = x 4 x 24. f(x) = x 3 + 2x 25. f(x) = x 6 + x

8 Algebraically determine whether the function is even, odd, or neither. (Non-Graphing Calculator) 26. f(x) = x 3 2x 27. f(x) = 4x 3 3x The graph of a function defined for x 0 is given. Complete the graph for x < 0 to make a) an even function and b) an odd function. (CALCULATOR) a. b. Find the inverse of the function. (CALCULATOR) 29. f(x) = 2x f(x) = 3 x 5

9 UNIT 3 Graphs of Functions This entire unit will be on the Non-Graphing Calculator portion of the exam! Suppose the graph of f is given. Describe how the graph of g could be obtained from f. 1. g(x) = 2x g(x) = x g(x) = 3 x 4. g(x) = 2x 5. g(x) = 1 2 (x + 3)2 6. g(x) = 1 3 x + 1 Write the equation for the graph of f with the given transformations. 7. f(x) = x 3 ; shift left 3 units, reflect over the x-axis, and shift down 1 unit. 8. f(x) = x ; shift right 2 units, stretch vertically by a factor of 3, and shift up 2 units. 9. f(x) = x; reflect over the y-axis and stretch horizontally by a factor of The graph of f is given. Sketch the 11. The graph of f is given. Sketch the graph of y = 2f(x + 1) 3. graph of y = f( 2x) + 1.

10 Graph the function. 12. f(x) = x 2 2x f(x) = x f(x) = x 4 + 8x 3 16x f(x) = (x + 3) 2 (x 5) 16. f(x) = x 2 4x f(x) = 3 x

11 18. f(x) = x 3 + 3x 2 9x f(x) = x 4 3x 3 9x x f(x) = (x + 2) 2 3 Find all the information from the equation to graph the rational function. 21. f(x) = x2 +x 12 x 2 +2x 8 Horizontal Asymptote: Slant Asymptote: y-intercept: Vertical Asymptote(s): Hole(s): x-intercept(s): Behavior near vertical asymptote(s):

12 Find all the information from the equation to graph the rational function. 22. f(x) = x2 +2x 8 x+3 Horizontal Asymptote: Slant Asymptote: y-intercept: Vertical Asymptote(s): Hole(s): x-intercept(s): Behavior near vertical asymptote(s): 23. Graph the piecewise defined function. 4 if x < 2 f(x) = { x 2 if 2 x 2 x + 4 if x > Write a piecewise defined function for the graph below.

13 CHAPTER 4 Exponential & Logarithmic Functions Find the domain of the function. (Non-Graphing Calculator) 1. y = log 3 (x 5) 2. y = ln(3 7x) 3. y = log 5 x 4 Graph the following functions. Identify the domain, range, and asymptote of the function. (Non-Graphing Calculator) 4. f(x) = 3 x f(x) = 2 x 3 D: R: HA: D: R: HA: 6. f(x) = log 2 x 7. f(x) = 3 + log 2 (x + 4) D: R: VA: D: R: VA: 8. Find the exponential function f(x) = a x 9. Find the logarithmic function f(x) = log a x whose graph is given. (CALCULATOR) whose graph is given. (CALCULATOR) 1 (2, 16 ) ( 1, 1) 2

14 Express the equation in exponential form. (CALCULATOR) 10. log 8 4 = log0.1 = ln(x 1) = 4 Express the equation in logarithmic form. (CALCULATOR) = = e x+1 = 0.5 Evaluate the expression. (Non-Graphing Calculator) 16. log log log log log e lnπ 22. log log log 2 6 log 2 15+log log Expand the expression. (CALCULATOR) 25. log 2 ( x2 yz 3) 26. ln 3r2 3 s Condense the expression. (CALCULATOR) 27. log a b + clog a d rlog a s (log 5x + 2log 5 y 4log 5 z)

15 Solve the equation. Find exact answers whenever possible; otherwise, round answers to four decimal places. (CALCULATOR) x = x 1 = e 12x = x = 6 x 33. log 4 2 = x 34. ln(2 + x) = log 2 (x 2 x 2) = log x + log(x 3) = log 5 x + log 5 (x + 1) = log 5 20 Applications. Formulas will NOT be given, so you may find it useful to include them on your notecard! (CALCULATOR) 38. Find the time required for an investment of $4000 to increase to $9000 if it is compounded quarterly at 6% annual interest. 39. Find the interest rate for an investment of $600 to triple if it is compounded continuously for 9 years?

16 40. A baseball card increased in value from $15 to $2000 in 25 years. Find its average annual rate of appreciation. 41. A man invests $6500 in an account that pays 6% interest per year, compounded continuously. a) What is the amount after 2 years? b) How long will it take for the amount to be $8000? 42. In 1990, a fish population in a man-made lake began with 150 fish. Since then, the population has had a relative growth rate of 9% per year. a) Write a function that models the amount of fish at time t, where t represents the number of years since b) What was the fish population in 2000? c) How long will it take the population to reach 2500 fish? 43. The half-life of radium-226 is 1600 years. How long will it take a 22-mg sample to decay to 18-mg?

17 Chapters 5 & 6 Trigonometric Functions of Angles 1. Convert 300 to radians. (CALCULATOR) 2. Convert 11π to degrees. (CALCULATOR) 3 For #3-7, find the value of x in the triangle. Number 7 must be exact. (CALCULATOR) x 5. 68º 12 53º 25 x 36º x x 85 60º 65º 60º x 30º Applications. (CALCULATOR) 8. From the top of a 200 ft. lighthouse, the angle of depression to a ship in the ocean is 23º. How far is the ship from the base of the lighthouse?

18 9. To estimate the height of a mountain above a level plain, the angle of elevation to the top of the mountain is measured to be 32º. One thousand feet closer to the mountain along the plain, it is found that the angle of elevation is 35º. Estimate the height of the mountain. 10. Points A and B (on the same side of a tower) are 16 m apart. The angles of elevation of the top of a tower are 38 and 48 respectively. Find the tower s height. Find the sign of the expression if the terminal point determined by θ is in the given quadrant. (CALCULATOR) 11. cos θ cot θ; QII 12. csc θ tan θ; QIV From the information given, find the quadrant in which θ lies. (CALCULATOR) 13. sec θ > 0 and cscθ < tan θ > 0 and sec > 0 Find the value of the trigonometric functions of θ given the quadrant in which the terminal point lies. (CALCULATOR) 15. cos θ = 4 ; Quad II 16. tan θ = 1 ; Quad III 5 4

19 Find the missing coordinate of P, using the fact that P lies on the unit circle in the given quadrant. (CALCULATOR) 17. P ( 3, y); Quad III 18. P (x, 1 ); Quad II 19. P (x, 2 ); Quad IV Find a) the reference angle and b) the terminal point P(x, y) determined by the angle. (Non- Graphing Calculator) 20. 3π π π º Find the exact value of the trigonometric function. (Non-Graphing Calculator) 24. cos 5π sin cot ( 3π 2 ) 27. csc 7π tan sec 13π 4

20 Find the amplitude, period, vertical shift, and phase shift of the function. Then graph the function. (Non-Graphing Calculator) 30. y = 2 sin 2 (x π 4 ) 1 Amplitude: Period: Phase Shift: Vertical Shift: 31. Use the graph to write an equation of the form y = a cos b(θ c) + d. (Non- Graphing Calculator) 32. Use the graph to write an equation of the form y = a sin b(θ c) + d. (Non- Graphing Calculator)

21 Identify whether the graph represents y = tan x, y = cot x, y = csc x, or y = sec x. (NON- CALCULATOR) Find the EXACT value of each expression, if it is defined. (Non-Graphing Calculator) 37. sin cos 1 ( 3 2 ) 39. tan tan (sin ) 41. sin (tan 1 ) 42. csc 5 (cos 1 7 ) 25

22 43. Find the length of an arc that subtends a central angle of 45 in a circle with radius 10 m. (CALCULATOR) 44. Claire is riding a bicycle whose wheels are 28 in. in diameter. If the wheels rotate at 130 revolutions per minute (rpm), find the speed at which she is traveling in mi/h. (CALCULATOR) 45. Sasha rotates a stone in a 3 ft. long sling at the rate of 15 revolutions every 10 seconds. Find the linear and angular velocities of the stone. (CALCULATOR)

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