AP Calculus Summer Assignment - Part 1

Transcription

1 AP CALCULUS SUMMER ASSIGNMENT Linear Functions AP Calculus Summer Assignment - Part 1 Determine the equation of the line that passes through the given points 1. (0,-1) and (5,9) 2. (-2,-1) and (1,5) 3. (3,6) and (3, -4) 4. (1,2) and (8,2) Write an equation through the given point that is (a) Parallel to the given line (b) Perpendicular to the given line Point Line 5. (2,1) 4x - 2y = 3 6. (-3,2) x + y = 7 Find a relationship (equations) between x and y such that (x,y) is equidistant from the two points. 7. (4,-1) and (-2,3) 8. (3,-2) and (-7, 1)

2 Piecewise functions: 2x 1, if x 1 g(x) = { 3x + 1, if x > 1 1. Evaluate g(5) 2. Evaluate g(1) 3. Graph the given function f(x) = { 3 2 x 1, if x < 2 x + 1, if 2 x 1 3, if x > 1

3 Quadratics: Converting from standard form to vertex form 1. f(x) = x 2 + 4x f(x) = x 2 18x + 86 Solving quadratics by factoring 3. x 2 8x + 12 = 0 4. x 2 49 = 0 5. x 2 11x + 30 = x + 28 = x 2 Solving quadratics using square roots 7. 4(x 1) 2 = (x 4) 2 18 = (x + 2) 2 5 = 8 Use the quadratic formula to solve the following equations 10. x 2 + 3x = x 2 18x = 12x x 2 + 4x = For the function f(x) = x 2 + 7, evaluate each of the following (a) f(3a) (b) f(b 1) (c) f(x) f(1) x 1 (d) f(x+h) f(x) h

4 Polynomials Graph the following polynomials (include any zeros and y-intercepts); state the domain and range 1. f(x) = x 3 4x 2 11x g(x) = 2x 4 x 3 42x Determine the points of intersection of the functions provided. 3. x + y =2 2x y =1 4. x 2 + y = 6 x + y = 4 5. x 2 + y 2 = 25 2x + y = y = x 3 4x y = -(x + 2)

5 Exponential and Logarithmic Functions Graph each function then state the domain and range 1. f(x) = ( 1 8 )x 2. g(x) = ( 1 3 )x 4 3. h(x) = 2e x Expanding and condensing logarithmic expressions Expand each logarithm 4. log(3xy) 5. ln(10x 3 y) 6. log 3y x 5 Condense each expression 7. 3log4 + log ln3 + 5ln2 ln8 Solving exponential and logarithmic equations Solve the following equations 9. 5 x = e 3x = log(2x 5) = lnx + ln(x + 2) = 3

6 Rational Functions Graph the functions. State the domain and range, label any vertical asymptotes. 1. f(x) = 4 x 3 2. g(x) = 4x2 x 1 3. s(t) = 3t 2 t 4

7 Solving Rational Equations 4. 2x 9 = 2 x x 5 x 5 x x = 4 x+1 x x = x+1 x For the function f(x) = 1/x evaluate the following: f(1 + h) f(1) h

8 Trig Functions Convert each angle to radian measure (express each one in terms of π). Do not use a calculator o o o o o Convert each angle to degree measure. Do not use a calculator. 6. 3π 2 7. π π A right triangle as an acute angle,, such that cos 3 5. Find sin. 10. A right triangle as an acute angle,, such that 12 sin. Find cot. 13 Without using a calculator evaluate the trig function at the given angle. o 11. cos 60 o 12. tan sin 120 o

9 14. You are standing 75 meters from the base of a building. You estimate the angle of elevation to the top of the building is 80 o. What is the approximate height of the building? Solve for (0 2π): sin cos 2 sin 2 2 Solve the given equations 18. arcsin(3x π) = ½ 19. arctan(2x 5) = -1

10 AP CALCULUS SUMMER ASSIGNMENT AP Calculus Summer Assignment - Part 2 1. You need to build a cylindrical water tank using 100 cubic feet of concrete. The sides and the base of the tank must be one foot thick. What are the dimensions that will maximize the capacity of the tank? Explain your solution and include any models/diagrams you use.

11 2. While camping, you and a friend filter river water into two cylindrical containers with the radii and heights given in the chart below. You then use these containers to fill the water cooler with the given dimensions. Radius Height (in.) (in.) Container Container Cooler 12 5 (a) Find the volumes of the containers and the cooler in cubic inches. (b) Using your results from part (a), write and graph an inequality describing how many times the containers can be filled and emptied into the water cooler without the cooler overflowing. (c) Convert the volumes from part (a) to gallons. Then rewrite the inequality from part (b) in terms of these converted volumes. (d) Graph the inequality from part (c). Compare the graph with your graph from part (b), explain why the results make sense.

12 3. Adam is at a lake, taking his dog for a walk. He throws a tennis ball from point A along the water s edge to point B in the water. His dog, Riley, first runs along the beach from point A to point D and then swims to fetch the ball at point B. (a) Riley s running speed is about 6.4 meters per second. Write a function, r(x), for the time he spends running from point A to point D. Riley s swimming speed is about 0.9 meters per second. Write a function, s(x), for the time he spends swimming from point D to point B. (b) Write a function, t(x), that represents the total time Riley spends traveling from point A to point D to point B. (c) Graph the function(x) from part (b). (d) Determine the value of x that minimizes the amount of time it takes for Riley to get the ball. Explain your solution. (e) What is the total distance Riley covers include the distance run on land and the distance he swam in order to retrieve the ball? Explain your solutions.

13 4. You have 100 meters of fencing material to enclose a rectangular plot. Your goal is to determine the dimensions of the plot such that you enclose the maximum area possible. (a) Express the area, A(x), of the rectangular plot as a function of the length, x, of one side. (b) Analyze the problem numerically by completing the table x A(x) According to this table, what do you think the dimensions of the plot should be to enclose the maximum area? Explain. (c) Use a graphing utility to graph the area function. What is the domain of the function? According to your graph, what are the dimensions of the plot that would yield a maximum area? (d) Solve the problem algebraically by putting A(x) into vertex form. How does this form of the function allow you to find the dimensions that produce a maximum area? (e) Discuss the strengths and weaknesses of the three strategies used in parts (b), (c), and (d). (f) Suppose you were not restricted to a rectangular plot. Would you be able to use 100 meters of fencing to enclose a greater area? (g) Suppose the rectangular plot runs along a building so that you need to fence only three sides. What dimensions will now yield a maximum area with 100 meters of fencing? Explain your method of determining these dimensions.

14 5. The following table represents the monthly precipitation p (in inches) for Seattle, Washington over the past year. In the table month t = 1 corresponds to January. t p (a) Create scatter plot of the data. (b) Determine a model (equation) that best fits the data. Explain why you chose this type of model and why you think it is a good fit. (c) Use your model to predict the amount of precipitation in December. (d) Use your model to predict the amount of precipitation in June of the following year.

15 6. Suppose you are driving from Atlanta to Miami. The trip is about 700 miles and takes you 12 hours. Your average speed is: Average speed = Distance time = miles per hour This concept can be generalized as follows. Let f be a function defined on the interval [a,b]. The average rate of change off from a to b is: This expression is called a difference quotient Average rate of change = f(b) f(a) b a (e) Calculate the average rate of change of f(x) = 3x + 4 on the interval [2,6]. Select any other interval and show that you obtain the same average rate of change. Explain why you obtain the same average rate of change. (f) Calculate the average rates of change of f(x) = x2 on the intervals [1,3] and [4,6]. Are they equal? Explain. (g) Let f(x) = x2 2. I. Calculate the average rates of change of f on the intervals [1,3], [1,2], [1,1.5] and [1,1.1]. II. Find an expression for the average rate of change of f on the interval [1,1+h], where h is any real number. III. Create a table that shows the average rate of change of f for several values of h. Choose values of h that get closer and closer to 0. What value does the average rate of change of f approach as h approaches 0? IV. The answer to part (b) is the slope of the tangent line to the graph of f at point (1,f(1)). Graph this line and f on the same graph. Describe the behavior of the graphs near the point (1,f(1)).

16 7. (a) Use a graphing utility to graph the function f(x) = sinx. On a separate sheet of paper, plot your estimates of the slope of this curve at various x values. For instance the slope is approximately 1 at x = 0, 0.8 at x = 0.5 and 0 at x = 1.5. After you have plotted 15 or 20 points, connect them with a continuous curve. Do you recognize this curve? (b) The slope of the tangent line was given by the formula on page 828. The difference quotient f(x+h) f(x) will be a good approximation of the slope if h is small. If you select h = 0.01, you h have the following approximation for the slope of the sine curve at x. sin(x ) sinx m 0.01 Complete the table below, plot these points on a graph and then compare your results with part (a) x m (c) To calculate the slope of the tangent line to f(x) = sinx algebraically, you need two trigonometric limits: sin (h) Limit 1: lim = 1 h 0 h 1 cos (h) Limit 2: lim = 0 h 0 h Use these limits to find a formula for the slope of the sine curve.