Welcome to the Jeopardy of Calculus AB. The questions are based on Chapter 4 Materials. Let’s go to the Game Board Categories Area Integration Mental Funds U-Substitution Potpourri $ 100 $ 100 $ 100 $ 100 $ 100 $ 200 $ 200 $ 200 $ 200 $ 200 $ 300 $ 300 $ 300 $ 300 $ 300 $ 400 $ 400 $ 400 $ 400 $ 400 $ 500 $ 500 $ 500 $ 500 $ 500 Final Jeopardy Area 100 Find the sum. 20 2 i i 1 Answer Answer The distributive property of summation says that we can place any constant outside of the sigma notation. 20 2 i i 1 (20)(20 1) = 2 2 210 420 2 Categories Area 200 Find the sum. 4 1 2 k 0 k 1 Answer Answer We evaluate this sum by replacing k with 0, then 1,2,3, and 4 respectively. 1 1 1 1 1 158 2 2 2 2 0 1 1 1 2 1 3 1 4 1 85 Categories Daily Double!!! • Choose the amount you would like to wager. (Up to half your score) Question Area 300 Find a formula for the sum of n terms. Use the formula to find the limit as n infinity. n 16i lim x 2 i 1 n Answer Answer Evaluate the summation: After simplifications: And the answer: Categories 16 n n 1 lim x 2 n 2 1 8 lim x 1 lim x n 81 0 8 1 8 Area 400 Determine a value that best approximates the area of the x-axis and the graph of the function over the indicated interval. f(x)= 4 – x2, [ 0,2] Answer Answer First sketch a graph of the function: Then approximate counting the squares: A=1+1+1+1+0.5+0.5+0.5=5.5 Categories Area 500 Find the sum: 2i 1 2 i 1 n 1000 Answer Answer 1000 1 Solve the summation: 2i 1 1000 i 1 1000 2 1002 1.002 1000 1000 Categories Integration 100 x 4 x 2 xdx 3 Answer Answer Integrate each part separately. Don’t forget the C! Then rewrite to: 3x 4 C 2 Categories Integration 200 3dx Answer Answer Integrate 3x C Categories Integration 300 ( t sin t ) dt 2 Answer Answer First, integrate t2 separately from –sint. Then integrate –sint. (Chain Rule) Then rewrite to: 2t cos t C Categories Integration 400 Solve the differential equation: f (s) 6s 8s , f (2) 3 ' Answer 3 Answer First, integrate the equation. Then plug in 2 for x, and find C. Then rewrite to: f (2) 3s 4s 23 2 Categories 4 Integration 500 Calculate the indefinite integral: 5x 4 x 2 x dx x 4 Answer 2 Answer First divide the entire equation by the x in the denominator. Then integrate each piece independently of each other. 5 4 2 x 2x 2x C 4 Categories Mental Funds 100 Find the area of the region bounded by the equation: y 2 x 3x 2 2 On the interval [0,2]. Answer Answer 2 3 3 2 2 x x 2x 0 3 2 2 3 3 2 10 (2) (2) 2(2) 3 2 3 Categories Mental Funds 200 Find the average value of: f ( x ) 3x 2 x 2 On the interval [1,4]. Answer Answer 1 4 2 1 3 2 4 3x 2 xdx = x x 1 3 1 3 1 43 42 13 12 3 48 16 3 Categories Mental Funds 300 Evaluate the definite integral: 1 0 Answer (2t 1) dt 2 Answer 1 First Expand: 0 (4t 2 4t 1)dt 1 Then Integrate: 4 3 2 t 2 t t 3 0 Then plug in the 4 3 Numbers: (1) 2(1)2 (1) 3 Answer: Categories 1 3 Mental Funds 400 Using the second fundamental theorem of Calculus find F’(x): F x t 2 dt 2 Answer x Answer Simply plug in x for t. F '( x) x 2x 2 Categories Mental Funds 500 Using the second fundamental theorem of Calculus find F’(X) x3 F x sin t 2dt 0 Answer Answer Using chain rule derive the inside and then plug in the 3 x. F '( x) sin( x ) (3x ) 3x sin x 3 2 Categories 2 2 6 U-Substitution 100 2sin x cos xdx Answer Answer sin 2xdx Thus: u = 2x; du = 2dx; ½du = dx ½(cosu) ½cos2x ½cos2x Categories U-Substitution 200 x dx x2 1 Answer Answer U = x2+1; du = 2x dx; ½du = x dx du ½ u =½ Categories U-Substitution 300 cos x dx sin x Answer Answer x 1, x 3 Categories U-Substitution 400 sec x tan x dx sec x 1 Answer Answer Categories y 2x 1 y 3 x2 U-Substitution 500 2 csc x dx cot 3 x Answer Answer y ( x 2) 3 2 Categories Potpourri 100 Derive: (2x 3) (5x 3x 5) 2 Answer 2 Answer (4 x 0) (10 x 3 0) 6 x 3 3(2 x 1) Categories Potpourri 200 Find the limit: sin x lim x x 0 Answer Answer You should know that as x approaches 0 in the limit: sin x lim x x 0 The limit equals one. Categories Potpourri 300 Derive this equation: y 2xy 2x 4 2 Answer 2 Answer 2 yy 2( xy y(1)) 4 x 0 ' ' 2 yy 2xy 2 y 4x 0 ' ' 4 x 2 y 2 x y y 2 y 2x yx ' Categories Potpourri 400 Integrate: 3 x Answer 2 x 2dx 3 Answer ( x 2) ( x 2) (3x )dx 3/ 2 3 3 1/ 2 Categories 2 3/ 2 2 3 C ( x 2)3/ 2 C 3 Potpourri 500 Integrate the indefinite Integral: 2 t ( t ) dt t 2 Answer Answer Expand: (t 3 2t )dt 2 Integrate: 1 t 4 2t C 1 t 4 t 2 C 4 2 4 1 Answer: t 4 t 2 C 4 Categories Final Jeopardy •Think about your wager for final Jeopardy! • The category is: The Angle between two Vectors! •Just Kidding! • Your Category is: Area Approximations! Decide wager amount Wager some, all, or nothing… Use Trapezoid Rule to Approximate: n=4 1 0 1 x3 1 Solution: Solution: We now use Trapezoid Rule to approximate the definite integral. 2 1 f (1) 2 f (1.25) 2 f (1.5) 2 f (1.75) f (2) 8 Answer: [2.0599] Area! • Using integration, and arithmetic sums, to determine the area produced between the graph and the x-axis. Categories Integration! • Using basic integration to determine both definite and indefinite integrals. Don’t forget the + C! Categories Mental Funds! • Understand the fundamental theorems of calculus for this part…you’ll need them. TRUST ME! Categories U-Sub! • Using U-Substitution, find the integrals of some problems. • Warning…some of these problems are rather difficult! Categories Potpourri! • Anything and Everything From the Three Previous Chapters! HAHAHAHA! • Solvers Beware! Categories