f exist? Why or why not? Non-AP Calculus Summer Assignment 1. Use the graph at the right to answer the questions below. a. Find f (0).

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1 1. Use the graph at the right to answer the questions below a. Find f (0). b. On what intervals is f( x) increasing? c. Find x so that f( x). d. Find the zeros of f( x ). e. Approximate any value(s) of x such that f x x. f. Find f ( f (9)). g. Find the average rate of change of f from x to x 7. h. Does 1 f exist? Why or why not?. Let f ( t) t 7t 5. Find and simplify f ( t h) f ( t). h

2 f x x 1 x (calculator). Let a. Approximate all local maximum and minimum values of f rounded to decimal places. b. Determine all the zeros of f. Give exact values. t 1 4. Let f t and gt. Find and simplify f t t 7 g. 5. Let x 7 1 f ( x). Determine the y-intercept of f. x The volume of a right circular cone is given by the formula V r h. If the volume of a right circular cone is 50 cubic centimeters, express the radius as a function of height. 7. An airplane manufacturer can produce up to 15 planes per month. The profit made from the sale of these planes can be modeled by Px ( ) 0.x 4x where Px ( ) is the profit in hundred thousand of dollars per month and x is the number of planes made and sold. Based on this model, how many planes should be made and sold to maximize the profit? What is the maximum profit? 8. Let x 5 f ( x). ( Bx 1)( x ) a. Determine the value of B so that 1 y is a horizontal asymptote of f (x). 4 b. Determine the value of B so that f (x) has a slant asymptote. Find the equation of the slant asymptote.

3 9. Let x 7 f ( x) x c Determine the value of c so that x 9 is a vertical asymptote of f (x). 10. Determine a possible formula for the polynomial function y g(x) whose graph is shown below. Leave your answer in factored form. g. Determine the value of k so that 11. Let x k x k x 5x 1 of g x. x is a factor 1. Simplify the following trigonometric expressions: a. cos tan cot b. 1 sin t cost cost 1 sin t

4 c. tan w tan w 1 sec w 1. Solve for the indicated variable: x 4 a. y, for x 5 7x b c, for c 14. Let f t t 7 6 log a. Determine the asymptote of f. b. Find the x- intercept of f. 15. Determine the domain of the following functions. a. g( x) 1 x x b. h ( t) t 1 c. r ( w) w 7w 9

5 16. Solve each of the following equations for y, and determine if the equations represent functions of x. a. b. c. 17. Write a function for each of the following, using appropriate variable notation. Interpret the x and y intercepts, if applicable. a) A machine with an initial value of $5,000 depreciates in value by $1000 per year. Let the equation be the value after t years. b) A boy who is 4 feet tall is growing at a rate of inches per year. Let the equation be the height of the boy after t years. c) A person is paying $10 a week to a friend in order to pay back a $100 loan. Let the equation be the amount the person still owes his friend after w weeks. 18. A rental car company charges $9 for a car, plus $.0 per mile driven. Let the equation be the amount the customer pays when m miles are driven. 19. For the following functions, determine the zero(s), as well as the interval(s) for which the decreasing a) f x x x x 1 b) f t t t 1 t c) g r r r 4

6 x 1 0. Let f( x) and gx ( ) 1 x x 1. Find f ( g ( x )) and g( f( x )). Simplify completely. 1. A manufacturer makes t-shirts and has found that if they charge $0.00 per t-shirt, they will sell 900 t-shirts every week. If they drop the price to $17.00 per t-shirt, they will sell 1100 t-shirts. a. If the quantity of t-shirts sold is a function of the price in which they can sell the t-shirts, find the linear equation that models this problem. Use p to represent price and q f ( p) to represent quantity sold. b. Find and interpret f (15). c. Give a practical interpretation of the slope. d. Find both the x and y intercepts. e. Give practical interpretations of both of these intercepts.. Solve each quadratic equation algebraically: You should be using different techniques to solve these problems to be efficient. a. x b. m m c. p 16 p d. 40t 0.05t 0 e. y 5y 11 f. Solve for x : a x h k 0

7 . In winter with heavy snow falls, the animals in Yellowstone cannot freely roam to find food, so hay is dropped for them from helicopters. A helicopter 180 feet above the ground drops a bale of hay while the helicopter itself is raising feet per second. The height, s (t) (in feet) of the bale as a function of time is given by s ( t) 16t t 180 where t is the time since drop off (in seconds). a) What is the height of the bale seconds after it is released? b) How many seconds will it take the bale to reach the ground? c) When does the bale reach a maximum height? What is the maximum height? d) Sketch a graph of the height of the bale as a function of time. Explain the shape of the graph. 4. A motorcycle stunt rider jumped across Snake River. The path of his motorcycle was given approximately by H 0.005x.9x 600 where H was measured in feet above the river and x was the distance from his launch ramp. a) What was the rider s maximum height above the river? b) How far (horizontal distance) was the rider from the ramp when he reached his maximum height? c) How high above the river was the launch ramp? 5. Find an equation for each of the polynomial functions represented below. Leave your answer in factored form.

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11 6. Find the domain for each of the following functions. In part c., assume A and B are constants. a. ) ( 4 ) ( m m m m T b ) ( t t t t t f c. B Ar r r g 6 ) ( d. y y y y y h ) (

12 7. Give exact answers for parts (a) (g) for the following 4 functions. Be sure to support your answers. (a) (b) (c) (d) (e) (f) (g) Determine the equation of any horizontal or slant asymptote. Determine the equation(s) of any vertical asymptote(s). Give the y-intercept. Calculate the zeros of the function. When the input approaches infinity, is the function above or below the horizontal asymptote? When the input approaches negative infinity, is the function above or below the horizontal asymptote? Without using a calculator, neatly graph the function showing each asymptote with a dotted line. 1. f x x) x ( x 15 x. t 8 g ( t) t 16. w w 6w h ( w) w 1 4. x k x) x ( 1x 4 x 0 8. Solve for x : x e x 5xe x 0

13 9. Solve for t without using logarithms: 1 4 5t t 0. Write a possible exponential function in the form x y Cb, with b 0, b 1. b) (0, 5) and (, 10) 1. If $10,000 is invested at an interest rate of % per year, compounded semiannually, find the value of the investment if it was invested for 4 years.. If $10,000 is invested at an interest rate of % per year, compounded continuously, find the value of the investment if it was invested for 7 years.

14 . Expand each logarithm completely: 1. log 11yz 1 / 5x. ln t zw. log 5t x 4. 8x k ln 4 tr / 4. Rewrite as a single logarithm: a) 1 ln( 6x) ln x ln( x) b) log( 5z) log( x) log( y) log t 7. The following pairs of expressions are not equal. Explain/show why not. 5. Solve each of the following exactly and then approximate the solution using your calculator: 1. t t. z 4( ) 5 z 4. x 7 x 1

15 5. q q log( u 7) 6 7. log z log( z 15) 1 8. ln t t ln t 9. ln( x x) ln( x ) w w ln x 5ln x 6. Fill in the missing values in the chart below, where represents the distance on the unit circle, where 0 t. Negative means the algebraic sign of the value is negative. Give exact values. t cos t sin t tan t radians negative 5 negative undefined

16 7. Find the exact values of all 6 trigonometric functions for each value of t. If a value does not exist, write undefined. a. b. c. 7 t 4 5 t 6 t 8. Find the exact values of all 6 trigonometric functions given the following information: a. tan t 4 and the terminal point determined by t is in Q III. b. cos t A, where A 0 and the terminal point determined by t is in QIV. 9. Let v. Evaluate each of the following. Exact answers are required, not calculator 4 approximations. v cos b. cos a. v c. cosv d. cos v e. cosv f. cos v 40.. Fill in the following chart. Show appropriate work below. (Give exact answers and then round to the nearest degree or nearest hundredth of a radian.) Degree Radians 4 0

17 41. Solve for the given angle. Express your answer in radian form and assume 0angle. In most cases there are two angles. a. sin b. tan 0 c. sec is undefined d. cos x e. csc x 1 f. tan y 1 4. Find the angle in degrees. Round your answer to decimal places Find the exact value of each expression, if it is defined. 1 a. cos 1 b. sin 1 c. tan 1 1 d. cos cos e. cos 1 cos f. sin 1 cos 44. Simplify the following expressions: a. sin t cos t sin t cost b. sec xcsc x cot x tan x

18 tan A tan A c. sec A 1 sec A a) Find domain and range: b) Find domain and range: 46. For problems 1-5, find the solutions in the interval 0,. Give exact answers. 1. 4sin x 1 (NO calculator). sin x cos x sin x 0 (NO calculator). cos x 0. 8 (calculator) 4. cos x cos x 1 0 (NO calculator) 5. sin x cos x (NO calculator) 47. Suppose the population of a town is,000 people. If the population grows at a relative rate of 5.4% per year, a. Find a model giving the population after t years. b. How many years will it take for the population to increase by 80%?

19 48. Solve for the indicated variable. Give exact answers. a. log( y 1) log( y 1) log 8 (5x) b. e 9 1 c. ln( 7w ) d. log t log t 49. Find α and x. Approximate your answers to the nearest hundredth. 15 cm 0 cm x 50. Suppose 7 tan, where terminates in Quadrant II. a. Find cos. Give an exact answer. b. Find cos( ). Give an exact answer.

20 51. Solve algebraically (*no calculator) x + x - 11x < 1 5. Sketch by hand. a) y sin 4x b) y 6cos( x )