AP Calculus. Worksheet All work must be shown in this course for full credit. Unsupported answers may receive NO credit.. Write the equation of the line that goes through the points ( 3, 7) and (4, 5) in a) point-slope form b) slope-intercept form c) standard form. Write an equation of the line that passes through the point (, 5) and a) is parallel to the -ais b) is parallel to the y-ais 3. Find an equation of the line tangent to a circle with radius 5 and center (0,0) at the point (3, 4). Hint: A Tangent Line to a circle is always perpendicular to the radius at the point of tangency. 4. For what value of k are the two lines + ky = 3 and + y = a) parallel b) perpendicular
5. The relationship between Fahrenheit and Celsius temperatures in linear. a) Using the fact that water freezes at 0 C or 3 F, and water boils at 00 C or F (not your recollection of temperature formulas) find a linear equation that relates Celsius and Fahrenheit. b) Using your equation, find the Celsius equivalent of 80 F and the Fahrenheit equivalent of 0 C. c) Is there a temperature at which a Fahrenheit and a Celsius thermometer give the same reading? If so, what is it? 6. The pressure p eperienced by a diver under water is related to the diver s depth d by an equation of the form p = kd +, where k is a constant. When d = 0 meters, the pressure is atmosphere. The pressure at 00 meters is 0.94 atmospheres. Find the pressure at 50 meters. 7. Consider the point P (, ) and the line L: 3 + 5y =. a) Find the slope of L. b) Write an equation for the line through P and parallel to L. c) Write an equation for the line through P and perpendicular to L. d) What is the -intercept of L?
AP Calculus. Worksheet All work must be shown in this course for full credit. Unsupported answers may receive NO credit.. Piecewise Functions: Complete the following tetbook problems: page 0 #3, 33, 43, 45. A student begins saving money by hiding $50 they received for their birthday in an envelope in their bedroom. After forgetting about the envelope for 3 months, the student starts putting $0 a month into the envelope for the net 3 months. The student then gets a job and decides to increase the amount to $30 per month for the net 6 months. Write a piecewise function that models the amount of money in the envelope as a function of the number of months since their birthday, m. 3. Composite Functions: Complete the following tetbook problems: page 0 #5, 53 For questions 4 and 5, find f ( g( ) ) and determine the domain. 4 + 3 4. f ( ) = + 7 and g( ) = + 4 5. f ( ) = and g( ) - 3 = + - 4 6. Prove whether the following functions are even, odd, or neither. a) y = 3- b) g( ) - = c) ( ) 3 f = - 5 3
7. Complete the following tetbook problems: page #67 and 70 8. Write the equation of the following: a) the area A of a circle as a function of its diameter d. b) the area A of an equilateral triangle as a function of its side length s. c) the area A of a rectangle as a function of its width W, where the length L is twice as long as its width W. 9. Complete the following tetbook problems: page 0 #55 and 56 0. What are the three domain issues you must remember for this course?. What is the domain of the following functions? log ( -5) a) g( ) = -8 b) f ( ) = 9-. Find the domain and range of the following parent functions: (write your answers in interval notation) a) linear b) quadratic c) logarithmic d) sine e) inverse sine f) inverse cosine g) inverse tangent h) inverse linear (/) i) inverse quadratic (/ )
AP Calculus.3 Worksheet All work must be shown in this course for full credit. Unsupported answers may receive NO credit.. Complete the following from the tetbook: page 6 # - 4, 3-8. You buy a brand new car for $35,000 and find out it depreciates at.5% per year. Write an eponential equation modeling this situation. How much will your car be worth in 5 years? 3. The half life of Ra 6 is,60 years. If there are 0g initially, how much Ra 6 is left after,000 years? 4, The number of United States citizens y (in millions) who traveled to foreign countries in the years 988 through 996 are shown in the table below., where t = 8 represents the year 988. t 8 9 0 3 4 5 6 y 40.7 4. 44.6 4.6 43.9 44.4 46.5 50.8 5.3 a) Use the regression capabilities of your graphing calculator to find an eponential model that fits the data. b) According to the model, is the number of travelers increasing or decreasing? At what rate? c) Using your model, how many travelers were there in 980? 974? 006? d) Why is it important to let t = 8 represent the year 988? [Try answering question c using the actual year.] 5. Without a calculator, evaluate the epression 3 3 6- when = 9.
6. Solve the following equation with and without a calculator: 6-3 5 =. 7. Using your graphing calculator, let Y = and Y =. Graph both equations in the same window. there? a) Solve the equation = using your graphing calculator. Where are the solutions to this equation and how many are b) Clear the two graphs from the screen and use the equation - = 0. Solve for by graphing the left side of this equation. Where are the solutions to this equation and how many are there? c) What did you learn from the last two questions? The Number e Many eponential functions in the real world (ones that grow/decay on a continuous basis) are modeled using the base of e. Just like p» 3.4, we say.78 + e e». We can also define e using the function ( + ). As, ( ) 8. Complete the following table using the TABLE function of your calculator to help. Compare your results to n n 3 n ( + ) ( + ) ( + ) 0 00,000 0,000 00,000,000,000 0,000,000 00,000,000 r 9. Using the results from the table above, make a conjecture as to what ( ) n n + approaches as. n 3 ee,, and e. n
æ r ö 0. The formula for compound interest is given by B= Pç +, where B = ending balance, r = the annual interest rate, n = the çè n ø number of times each year the interest is compounded, and t = the number of years. Using your conjecture from question 8, if the interest is compounded continuously ( n ), what does this formula look like? nt. Suppose you invest $,000 in an account that earns you 5% annual interest. a) If the interest is compounded weekly, how much will you have in the account after 3 years? b) If the interest is compounded continuously, how much will you have in the account after 3 years?
AP Calculus Parent Function & Transformations Worksheet All work must be shown in this course for full credit. Unsupported answers may receive NO credit.. Given the function g () as shown to the right, describe the transformation, then graph and label the following: [Use another sheet of graph paper] a) y = g( ) - b) y = g( - ) c) y3 = g( + 5) d) y = g( - ) 4 e) y =- g( ) 5 f) y = g( ) 6 3. Complete the following questions from the tetbook: page 9: #7, 8,, 4 3. First graph f ( ) log3 =. Then graph the following transformations: a) y= 4- f ( ) b) y=-f ( - ) c) y= f ( + ) y d) y=-f ( - ) + 3 4 y 4. For a c, graph f ( ) = first, and then graph each transformation. a) y= + y b) y=- - 3 y
c) y= 3+ - y 5. Without using your grapher, state the parent function, describe the transformation, and sketch the following: [Use another sheet of graph paper] p a) y= log( -) b) y= 3sin( + ) c) ( ) - y=- + - d) y = e) y=- cos( ) + f) y= tan( - ) g) y = - ( + 3) h) y = + 3 ( -) i) y= ( -3) 3 - j) y= 3( + ) - k) = - 0 + 6 l) y=- 3- + 4 y 3 m) y 8 = + n) = ln(4)
AP Calculus Introduction to Conics Worksheet All work must be shown in this course for full credit. Unsupported answers may receive NO credit.. Eplain why the circle, parabola, ellipse and hyperbola are referred to as conic sections.. Graph each of the following equations: a) 4 y 3 b) 5 y 9 6 4 5 c) y 4 3 d) 8 y 3 3 e) 9y 4 36 y= 6- + f) ( ) 3. Graph the equation y y 6 8 0. [Hint: Use Completing the Square with the y s to write in a form that is familiar]
4. For each of the following use the given description to i) write an equation for the conic ii) graph the given conic a) circle, centered at (4, ), radius of b) ellipse, centered at ( 3,5), stretched units horizontally and 5 units vertically c) hyperbola that opens up and down, centered at (7, ), stretched 3 units vertically and 4 units horizontally. 5. Each of the following give equations for conic sections and tell how the conic is translated. i) Write an equation for the NEW conic ii) Graph the NEW conic a) Translate y - = up units and left 3 units. 6 5 b) Translate 6 + 9y = 44 down 3 units and right 4 units.
AP Calculus.5 Worksheet All work must be shown in this course for full credit. Unsupported answers may receive NO credit.. Eplain how to FIND an inverse of a given function algebraically.. Eplain how to VERIFY/PROVE two functions are inverses of each other. 3. Find the inverse function for each of the following functions and label them correctly. a) f ( ) = + 3 b) g( ) = 4 + 7-5 c) h( ) = + + 3 4. Verify your answers from question 3 are inverses. a) b) c)
5. Solve each of the equations below both algebraically AND graphically. a).045 t = b) 0.05t - e = 3 c) - 3 =- 6. Let f () = ln( ). a) What is the domain of f? b) What is the range of f? c) What are the -intercepts of the graph of f? d) Find f -. What is the domain and range of f -? e) Verify your answer to part d. t - - = + C 7 a) If W = 50 when t = 0, solve for C 7. Consider the equation: ln( 300 W) b) Using the value of C from the part a, solve for W in terms of t.
8. Write each of the following as an equation of y in terms of t. a) ln y t 4 = + b) ( ) ( ) ln y- - ln = t+ ln t 9. The half- life of a certain radioactive substance is 8 minutes. There are 5 grams initially present. a) Write an equation for the amount a present after t minutes. b) Using your equation from part a, algebraically determine when there will be one gram left. Verify graphically. 0. Prove that the function k( ) = - (for > 0) is its own inverse.
AP Calculus.6 Worksheet All work must be shown in this course for full credit. Unsupported answers may receive NO credit.. Graph the following trigonometric functions on a separate sheet of paper. For each graph, List the amplitude, period, phase shift, vertical shift, domain, and range. a) y= cos( 3- p) + b) y ( p) = 3sin + - p c) y= sin( + ) d) y ( p) 3 =- 3tan 3 + + p e) y = 3cos( ) f) y= sin( 4p) 6. Write both a sine AND a cosine function for each function shown. a) b) c) d)
3. Let f ( ) = - 3cos( ). a) What is the domain of f ()? b) What is the range of f ()? c) What is the period of f ()? d) Graph f (). Does it look like an even function, an odd function, or neither? Justify your response algebraically. e) Find all the zeros of f () on the interval é p, pù ë û. 4. Solve each equation in the specified interval both algebraically AND graphically. a) tan =.5, 0 p b) cos =- 0.7, p 4p c) csc =, 0< < p d) sin =- 0.5, - < < 5. Solve the following equations algebraically for the given variable (and interval if appropriate): a) Solve for t if 0 t p -t -t : e t t( e ) cos + sin - = 0 b) Solve for A: ( ) ( ) -t -t -t A - e cost + e cost- sint + e sint = 0
- 5 6. Given that q tan ( ) = -, find the values of the si trigonometric functions at the angle θ. 7. Without using a calculator, evaluate the following epressions: 7 a) sin( cos - 9 ( )) b) tan( sin - ( 3) ) 8. The following data represent the average monthly temperatures for Juneau, Alaska. (the graph is below) Month () Average Monthly Temperature ( F) (Jan) (Feb) 3 (Mar) 4 (Apr) 5 (May) 6 (Jun) 7 (Jul) 8 (Aug) 9 (Sep) 0 (Oct) (Nov) (Dec) 4. 8.4 3.7 39.7 47.0 53.0 56.0 55.0 49.4 4. 3.0 7. a) Using the graph, estimate a sine function that would fit the data. b) Your calculator is capable of finding a Sinusoidal Regression. Find the sine function that fits the data and compare it to your answer in part a. Avergae Monthly Temperature 60 50 40 30 0 0 0 0 5 0 5 Month ( = Jan) c) Using the equation in part b, what is the amplitude? Eplain the meaning of the amplitude in this contet. d) Using the equation in part b, what is the period? Eplain the meaning of the period in this contet. e) Using the equation in part b, what is the phase shift and the vertical shift? 9. Begin reviewing Chapter by completing the following questions from the tetbook: Page 56 # 43 odd, 53, 57 67.