AP Calculus BC Summer Review

Transcription

1 AP Calculus BC Summer Review Due September, 07 Name: All students entering AP Calculus BC are epected to be proficient in Pre-Calculus skills. To enhance your chances for success in this class, it is important that you refresh these topics. You need to show your work on separate sheets of paper. I will count the completion of this packet as homework grades for the first quarter. The material covered in this assignment will be tested during the st Quarter. Please send me an at Hoon.Kim@fcps.edu if you have any question.

2 Topic : Rational and Negative Eponents ** Simplify the epression. Leave your answer in positive eponents.. y y 5. sin. 5 ( ) ( 5) 5. 5 y y 6. ( ) Topic : Factor ** Factor completely y. 6y 08y. 9 8y 5. y y y 6. ( ) ( ) ( ) ( ) Topic : Rational Epression ** Simplify the epression ( ) 9 ( ). ( 9 ) ( 7 8) 5. h Topic : Comple Fraction ** Simplify the epression.. y y... y y y y

3 Topic 5: Solving Polynomial Equations ** Solve each equation =. 6= 0. ( ) 5( ) =. 9 8= 0 5. ( )( ) = 6. 7 = 7 Topic 6: Solving Rational Equations ** Solve each equation.. =. 6 6 =. = = 5 5. = = Topic 7: Logarithmic Function ** Simplify the epression.. log8. 6 ln 5 e. ln 8 ln ** Epand the epression using the property of logarithms.. y log 5 z. ln 5 ( ) 7 5. y log 5z ** Condense the epression using the property of logarithms. log 5 log log 5log. ( ) ( ) ( ). ln ( ) ln ( ) ln ( 5) ** Solve the equation.. log ( 5) =. ( ) ( ) 8 log 5 log =. log ( ) = log. ln ( 5) = ln ( ) ln ( ) 9

4 Topic 8: Eponential Equation ** Solve the equation. Leave the answer in an eact form.. = 8. e =. e e = 0. = 5 5 Topic 9: Sequences and Series. Write a rule for the nth term of the sequence ,,,,... and find a Write a rule for the nth term of the sequence,, 8,6,... and find a 0. Use sigma (summation) notation to write Use sigma (summation) notation to write Use sigma (summation) notation to write and evaluate the sum: Use sigma (summation) notation to write and evaluate the sum: Determine whether each infinite geometric series converges or diverges. If it converges, find the sum. A. n n= 0 B. n 5 n= 0 C. n n= D. n= 9 5 n Topic 0: Composition of Functions ** Find two functions f and g such that h( ) = [ f g]( ). Neither function may be the identity function y =.. h( ) =. h( ) = 6 9. h( ) ( ) ( ) = 5 7 ** Evaluate each epression using the values in the table.. ( f g)( 9). ( g f )( ). ( f f )( ). ( g g)( 6) 0 f() g() 0 ** If f ( ) =, g( ) =, and h( ) =, find the composition of two functions and state the domain.. g f ( ). g h( ). f h( ). f f ( )

5 Topic : Solving Inequalities ** Solve each polynomial inequality > ** Solve each rational inequality.. > Topic : Even and Odd Functions ** Determine whether the graph of each equation is symmetric with respect to the -ais, y-ais, or the origin. 5. f ( ) = 7. f ( ) = 5. ( ) f = 5 ** Determine if each relation is even, odd, or neither.. f ( ) = e e. f ( ) = 5. y = 5 Topic : Inverse Functions. Find an inverse of ( ) f =. 5 = 5. Show that f ( ) = e and g( ) ln ( ) ** Evaluate each epression.. arcsin. sin. arccos. ( ) cos 5. arctan ( ) 6. tan ( ) 7. ( ) csc 8. sec ( ) 9. cot ( ) 0. arctan ( 0 )

6 Topic : Transformations of Functions ** Describe the transformations from f ( ) to g( ), where g( ) is defined below. g = f 5. ( ) 7. g( ) = f ( ). g( ) = f ( 5). g( ) = 9 f ( ) 5. g( ) = f ( ) 5 6. g( ) = f ( ) ** Describe the sequence of transformations from f ( ) to g( ). f ( ) = and g( ) = 5. f ( ) = and g( ) = 7 ** If y =, generate the function by following the transformations in that order.. Vertically stretched by a factor of, units to the left, 5 units down, and then reflected in the -ais.. Horizontally compressed by a factor of, reflection in y-ais, 7 units up, and then reflected in the -ais. f = ** Sketch the following graph using ( ). y f ( ) =. y = f. y = f ( ). y = f ( ) 5. y = f ( ) 6. y = f ( )

7 Topic 5: Function Analysis. Use the complete graph of a polynomial function f ( ) [,0] A. Is the degree of f ( ) even or odd? B. Is the leading coefficient of f ( ) positive or negative? C. What are the distinct real zeros of f ( )? D. What is the least degree of f ( )? and -scale is E. How many turning points does it have? F. Stationary inflection point occurs at what value of? G. Describe the end behaviors f. Use the graph of ( ) = 5 A. Find the domain B. Find the equation of vertical asymptote C. Find the equation of the horizontal asymptote D. Find the range E. Is it an even function or odd function? F. Describe the vertical asymptotic behaviors f ( ) = f ( ) =. Use the graph of f ( ) = 6 A. Find the domain. B. Find the equation of the vertical asymptote. C. Find equation of the horizontal asymptote. D. Find the range E. Is it an even function or odd function? F. Describe the vertical asymptotic behaviors f ( ) = f ( ) =

8 . Use the graph of f ( ) A. Rewrite f ( ) in a factored form. 6 = 5 B. Find the domain C. Find the equation of the vertical asymptote. D. Find the equation of the horizontal asymptote. E. Find the range. F. Find the -intercepts and y-intercept. 5. Use the graph of f ( ) = A. Find the domain B. Find the equation of the vertical asymptote. C. Find the -intercepts and y-intercept D. Find the equation of the slant asymptote. E. Find the range. 6. Graph and analyze f ( ) = e A. Parent function: B. Domain: C. Horizontal Asymptote: D. Describe the behavior near the horizontal asymptote using the it notation. E. Describe the transformations. F. Range: G. Describe the behavior of the of the function using the concavity. H. Key points: I. Graph the function.

9 Topic 6: Piecewise Functions ** Use f ( ), =, >. Evaluate the function. A. f ( ) B. f ( 0) C. f ( ) D. f ( 6). Graph the funciton.. Is the function continuous at =? ** Use f ( ) 6, < =,. Evaluate the function. A. f ( ) B. f ( ) C. f ( 0) D. f ( 5). Graph the funciton.. Is the function continuous at =? Topic 7: Trigonometry ** Simplify the epressions. b g. sin cot sin. sinθ cosθ sinθ cosθ θ θ θ. sin 85 sin 5 ** Verify each identity. tan cos = tan. tan cot = sec csc. ( )( ). cos sec tan =. sin cot sin = csc sin

10 ** Use the Sum and Difference Identities to find the eact value of each trigonometric epression. [ α β] = α β α β [ ] cos cos cos sin sin [ α β] = α β α β [ ] sin sin cos cos sin. sin ( 7 ) cos( ) cos( 7 ) sin ( ). cos α β = cosα cos β sinα sin β sin α β = sinα cos β cosα sin β cos π cos π sin π sin π 8 8. sin ( 0 ) cos( 80 ) cos( 0 ) sin ( 80 ). cos π cos π sin π sin π π cos sin 5. ( ) ** Use the Sum and Difference Identities to derive the Double-Angle Identities. cos θ = cos θ sin θ = sin θ = cos θ. sin ( θ) = sinθcosθ. ( ) cos θ = cos θ sin θ = sin θ = cos θ to derive the Power-Reducing Identities. ** Use ( ). cos sin =. cos cos = ** Use the Power-Reducing Identities sin cos cos =, cos = to derive the Half-Angle Identities.. cos sin = ±. cos cos = ± ** Find all solutions of the equation in the interval [ 0, π ]. sin cos = 0. sin cos cos sin = 0. sin cos = 0. tan = sin

11 Topic 8: Evaluate Limits ** Evaluate Limits Graphically.. Use the graph of f ( ) A. Evaluate f ( ) B. Evaluate f ( ) C. Evaluate ( ) f D. Evaluate f ( ) E. Is f ( ) continuous at =?. Use the graph of f ( ) A. Evaluate f ( ) B. Evaluate f ( ) C. Evaluate f ( ) D. Evaluate f ( ) E. Is f ( ) continuous at =? ** Evaluate Limits Analytically.. Find each it if f ( ) = and g ( ) = c c A. f( ) g ( ) c B. [ f( ) ] c C. f( ) g ( ) c g ( ) f ( ) D. [ f( ) ] c g ( ) ** Evaluate Limits Algebraically ( ) sin 0

12 ** Evaluate Limits algebraically or graphically (without using a calculator) ( csc ) 0. cos π ** Continuity. Find the value of k that makes the piecewise function f ( ) be continuous. k > for f( ) = for. Write down the piecewise function so that sin f( ) = is continuous for all real numbers.. Write down the piecewise function so that f( ) = is continuous for all real numbers

13 Topic 9: Derivative ** Definition of the Derivative ( ) f = h 0 ( ) ( ) f h f h. Find the equation of the tangent line to the graph of ( ). Find the equation of the tangent line to the graph of f ( ) = at ( ) f = 5 at = using the definition of the derivative. 7, 7 using the definition of the derivative.. Evaluate ( ) ( ) ( ) h 7 h h 0 h ** Find the derivative of each function.. f ( ) =. f ( ) =. f ( ) = ( ). f ( ) = 5. f ( ) = e 6. ( ) f = b log b f sin cos 7. ( ) = 8. f ( ) = π 9. ( ) f = ln e ** Application of Derivative. The positon d of the skydiver in feet relative to the ground can be defined by ( ) passed after the skydiver eited the plane. A. What is the average velocity of the skydiver between the nd and 5 th seconds of the jump? B. Find the equation for the instantaneous velocity vt ( ) of the skydiver. C. What is the instantaneous velocity of the sky diver at and 5 seconds? d t = 5, 00 6t, where t is seconds. Tourists standing on a 00-foot-tall viewing tower often drop coins into the fountain below. The height of coin falling from the tower after t seconds is given by ht ( ) = 00 6t. Find the instantaneous velocity v(t) of the coin at seconds.. Consider the quadratic model ( ) ht = 6t 60t 80 for the height, h in feet, of an object t seconds after the object has been projected straight up into the air. How long does it take to reach the maimum height? What is the maimum height? Note: You can solve this problem with or without using the calculus method. A rectangular plot is enclosed by 00 meters of fencing and has an area of A square meters. Show that the area is maimized when the rectangle is a square. Note: You can solve this problem with or without using the calculus method