Limits and Continuity. 2 lim. x x x 3. lim x. lim. sinq. 5. Find the horizontal asymptote (s) of. Summer Packet AP Calculus BC Page 4

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1 Limits and Continuity t+ 1. lim t - t + 4. lim x x x x x-. lim x 0 4-x- x 4. sinq lim - q q 5. Find the horizontal asymptote (s) of 7x-18 f ( x) = x+ 8 Summer Packet AP Calculus BC Page 4

2 6. x + 1 lim x-1 x For what value of a is the function justify your answer. ìx - x+ ï, if x¹ f ( x) = í x- ï îa, if x= continuous? Show your work to 8. Find the vertical asymptote(s) of f ( x) = - + x -1 x x 9. What does it mean for a function to be continuous at x = a? You may describe this in your own words, but also include mathematically sound justification. Identify the different types of discontinuity. Summer Packet AP Calculus BC Page 5

3 Differentiation 1. What is the slope of the tangent to the curve x + y =- 7, when x =?. Find dy dx, if x + y x - = +. ( x+ h) - ( x) ln ln lim h 0 h Use the table below for questions 4 and 5. x 1 f(x) 0 1 f (x) g(x) g (x) If hx ( ) = f ( xg ) ( x), find h () 5. If k( x) = g( f ( x)), find k() =, then f '( x ) = 6. If f ( x) 4sec ( 5x) Summer Packet AP Calculus BC Page 6

4 7. Find dy dx if x+ 7 y= 5 - x 8. Write the equation of the normal line to y x + 1 = when x = A particle moves along a line so that its position at any time t³ 0 is given by the function st ( ) = t - 4t+, where s is measured in meters and t is measured in seconds. a. Find the displacement of the particle during the first seconds. b. Find the average velocity of the particle during the first 4 seconds c. Find the instantaneous velocity of the particle when t = 4. d. Find the acceleration of the particle when t = 4 e. Describe the motion of the particle. When does it change directions? When is it speeding up? Summer Packet AP Calculus BC Page 7

5 Applications of Differentiation 1. At what value of x does the function y x e 4x = 1. - change concavity?. Find all critical points for the function f ( x) = x + 5x - 8x- 10 é pù. Let y= sinx on ê 0, ë ú. Approximate the value c guaranteed by the Mean Value Theorem. Please û show all work to justify your conclusion. 4. A farmer has 160 meters of fence to enclose a rectangular area against a straight river. He only needs to fence in three sides. What is the maximum area that he can enclose with his materials? 5. Write the equation of the tangent line and the normal line of f ( x) = x + x- 1 at the point (-1, 4). Summer Packet AP Calculus BC Page 8

6 6. Consider the function f ( x) = x - x 4 a. Find the intervals on which f(x) is increasing or decreasing. b. Locate all extrema. c. Find the intervals on which f(x) is concave up or down. d. Find all points of inflection. e. Sketch the graph of f(x). Summer Packet AP Calculus BC Page 9

7 Integration 4 1. Evaluate the integral ( - + 1) ò x x dx. ò cos( x) dx=. - + x 5x x 1 ò dx= dx 4. ò 5 x ln x = Summer Packet AP Calculus BC Page 10

8 5. ò x 1+ x dx 6. ò 0 x e dx 7. Find the total area of the region bounded by the curve = and the x-axis on [ 1,1] y x The rate at which water flows out of a pipe is given by a differentiable function R(t). The table below records the rate at 4-hour intervals for a 4-hour period. time (hours) R(t) (gallons per hour) a. Use the trapezoidal rule with 6 subintervals of equal length to approximateò Rt ( ) dt. 0 b. Explain the meaning of your answer in terms of the water flow, using correct units. Summer Packet AP Calculus BC Page 11

9 c. Is there a time between 0 and 4 such that R'( t ) = 0? Justify your answer. d. Suppose the rate of the water flow is approximated by Qt ( ) = 0.01( t- t ). Use Q (t) to approximate the average water flow during the 4-hour period. Indicate units of measure. x ( ) sin 9. If g x ( ) =ò t dt, then g ' ç æp ö = è 6ø p 10. The velocity of a particle moving along the x-axis is given by vt ( ) = t - 9t for t³ 0. a. How far from the origin is the particle when t = 5? b. How far has the particle traveled (in total) in the first 5 seconds? Summer Packet AP Calculus BC Page 1

10 Polar Equations and Parametric Equations 1. Change from polar to rectangular form. (-,p). Change from rectangular to polar form. æ 1 ö a. ç -, è ø b. ( 8, - 15). Change the rectangular equation to polar form. a. x y x = 0 Summer Packet AP Calculus BC Page 1

11 b. x 5y = 4. Convert from polar form to an equation in rectangular form. r= 4sinq 5. Obtain an equation in terms of x and y by eliminating the parameter. Then identify the curve. x= t a. y= t b. x= + sinq y= + cosq Summer Packet AP Calculus BC Page 14

12 Vectors and Dot Products 1. Given a= 5, - 1 and b= -,- 6. Find the following: a. a + b b. a + b c. a- b. Find a unit vector that has the same direction as - i+ 7j.. What is the angle between the given vector and the positive direction of the x-axis? i+ j Summer Packet AP Calculus BC Page 15

13 4. Determine whether the given vectors are orthogonal, parallel or neither. a= 4,6 b= -, 5. Find the angle between the vectors. Give your answer using an exact expression (no decimals). a= -,5 b= 5,1 Summer Packet AP Calculus BC Page 16

14 Sequences and Series 1. Write the first five terms of the sequence. æpnö a. an = sinç è ø 1a. b. b. a n æ 1ö = ç - è 4ø n 1b.. Name the next two apparent terms of the sequence. Describe the pattern you used to find these terms. a. 5,10,0,40,... a. b. b., -,,-,... b Simplify the ratio of factorials.. ( n- ) ( n+ ) 1! 1! Summer Packet AP Calculus BC Page 17

15 4. Find the sum, if possible. å 0.9 n 4a. - a. ( ) 1 n= 1 b. 50 å n= 1 5n+ 4b. 1 c. ( n - n+ ) å 4c. n= 1 Summer Packet AP Calculus BC Page 18

Chapter 10 Conics, Parametric Equations, and Polar Coordinates Conics and Calculus

Chapter 10 Conics, Parametric Equations, and Polar Coordinates 10.1 Conics and Calculus 1. Parabola A parabola is the set of all points x, y ( ) that are equidistant from a fixed line and a fixed point