AP Calculus BC Summer Packet 7 o The attached packet is required for all FHS students who took AP Calculus AB in 6-7 and will be continuing on to AP Calculus BC in 7-8. o It is to be turned in to your AP Calculus BC teacher on the first or second day of school. o You must show all work (either in the packet or on separate sheets of paper). Your work should be ORGANIZED and IN ORDER! o You can epect a test on this material at the end of the first week of school. Suggestions: o Don t do the packet right away. Wait until the middle of the summer the purpose is to refresh your memory so that I can avoid re-teaching the material. o If you get stuck, feel free to email Mrs. Kraemer at amanda.kraemer@ocps.net or search various online sources (YouTube has nice tutorials, as do Khan Academy and Brightstorm Math Google any of them) o You should be able to complete all problems WITHOUT the use of a graphing calculator (several eceptions are noted). However, feel free to use one to check your work. Have a nice summer!
Limits and Continuity + 7 +. lim = cos cos a a. lim = a + lim + 8. = 4 cos 5. lim = + 5 6. lim = 6 4. = lim sin 5 7. lim ( + ) 8 =, 8. Find a value of k that makes f ( ) = continuous at =. sin( k), > 9. If a function f is discontinuous at = 4, which of the following must be true? I. lim f ( ) does not eist 4 II. ( 4) f does not eist III. lim f ( ) lim f ( ) 4 4+ IV. f ( ) f ( 4) lim 4. For each part draw an eample of a function that satisfies the conditions: a. f ( ) eists but lim f ( ) does not eist. b. lim f ( ) eists but ( ) f does not eist.
Derivatives For eercises -5, find the derivative of the given function. + y = 4. y = + 4. y = 5 + 8 5. s = cos ( t). r = ln( cos ) 6. Find dy for y + + y = d 7. Find the equation of the tangent line to y = at = 8. Consider the function f ( ) = + 9 +. On what intervals is this function: a. Increasing b. Decreasing c. Concave Up d. Concave Down e. What are the local etrema for this function: f. What are the inflection points for this function?
9. Let f be a function with f '( ) = + and f ( ) = 6 a. Write the linear approimation (tangent line) of f at = b. Using your equation from part a, approimate f (.) c. Is part b is an under-approimation or over-approimation? (Hint: use f ). Suppose that f '( ) = 4, g '( ) =, f ( ) =, g ( ) =, f '( ) =, and g '( ) = 5 derivatives at = of the following functions: h = f g a. ( ) ( ( )). Find the b. k ( ) = g( f ( ) ) c. r ( ) = f ( ) g( ) d. ( ) s = f g ( ) ( ). The graphs (i), (ii) and (iii) given in the figure below are the graphs of a function f, and its first two derivatives, f and f, though not necessarily n that order. Identify which of these graphs is the graph of f, which is f, and which is f. Justify your answer. 4
. The graph of f ( ) is shown below. f ( ) -5-4 - - - 4 5 6 a. On what interval(s) is f( ) increasing (approimately)? Decreasing? Eplain. b. Find the - values of all critical points of f( ) and determine whether they are a maimum, minimum, or neither. c. On what interval(s) is f( ) concave up? Concave down? Eplain. d. Find the -values of the inflection point(s) of f( ).. If the figure shows the graph of f, and g( ) = f ( t) a. g ( ), g( ), g( ), and g( ) dt, find: y Graph of f (,) b. g '( ), g' ( ), g' ( ), and g' ( ) (, ) (, ) c. the interval(s) where g is increasing (, ) d. the interval(s) where g is concave up 5
Integrals For eercises 4-8, evaluate the integral. 4. 5d π 7. 4 d (Hint: interpret this as an area and use Geometry) 5. sec θd θ d 8. + cos d tdt 6 + 6. ( ) d 9. Compute the T 4 (trapezoidal with 4 subintervals) approimation for ( ) 4 d using subintervals of equal length. (Set-up is sufficient you do not need to simplify your answer.) 6
Note: You may use a calculator to evaluate the integrals you use on this page, but please show all set-up!. Find the area enclosed between y = 4 and y = 4. The base of a solid is the region enclosed between the graphs of y = sin and y = sin from = to = π. Each cross section perpendicular to the -ais is a semicircle with diameter connecting the two functions. Find the volume of the solid.. Find the volume of the solid generated by revolving the region bounded by the following functions about the -ais: y =, y =, =. Find the volume of the solid generated by revolving the region from # around the line y = 7
Review Polar Equations Convert the following polar coordinates to rectangular coordinates: 4. (i), 4 π (ii) (,π ) Convert the following rectangular coordinates to polar coordinates:, 5 5. (i) ( ) (ii) (, ) Review Summation Notation Epand the following: 4 n 6. n= 7. n= n Congratulations! You finished the Summer packet and are ready to start AP Calculus BC!!! 8