Section A Solving Absolute Value Inequalities

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1 AP Calculus BC Prerequisite Notes for Summer Review Packet for students who completed AP Calculus AB These notes are designed to help ou understand how to accuratel complete the Summer Review Packet. Go to m website click on AP Calculus BC and then click on Prerequisite Chapter. Under the heading Notes Videos, when ou click on the link for each Section, ou will be directed to m videos so ou can see and hear m eplanation of how to solve the eample problems in order to complete the notes in this packet. The first eam of the ear will be mostl based on these notes and the problems in the Summer Review Packet. If ou have questions as ou work on the packet, ou can me antime at toshdemse@cusd.com.. I will also have four optional sessions during the summer where I can help ou out. You can also attend one or more of the optional tutoring sessions during the summer that will take place in m classroom, D06, on Frida August 0 th from p.m. to p.m., on Saturda August th from p.m. to p.m., on Sunda August th from p.m. to p.m., and on Saturda August 8 th from p.m. to p.m. To receive tet messages from me via Remind, please send a tet message to 800 and tpe in the If ou have not previousl used Remind for another teacher, ou will receive a tet message back that requests that ou tpe in our name. You will earn 0 points of etra credit for signing up for Remind. Section A Solving Absolute Value Inequalities Recall: absolute value is the distance awa from zero on the number line. Steps For Solving Linear Absolute Value Inequalities ) Isolate the absolute value b getting the absolute value b itself on one side of the inequalit. ) Draw a sketch on the number line that represents the correct distance awa from zero. ) If the inequalit sign is < or, rewrite as a compound inequalit and solve b isolating the variable. ) If the inequalit sign is > or, rewrite as two inequalities and solve each b isolating the variable. 5) Graph the solution on the number line. Eamples: Solve the following inequalities and graph the solution. <

2 + < Steps For Solving Non-Linear Absolute Value Inequalities ) Isolate the absolute value b getting the absolute value b itself on one side of the inequalit. ) If the absolute value has no effect on the epression inside it, then simplif and get one side of the inequalit equal to zero. ) Factor completel and set each factor equal to zero. ) Solve each of the equations from step, and put the solutions on the number line. 5) Test a number on each interval between the solutions found in step in the factored form to determine whether the interval is positive or negative. 6) Graph the solution on the number line based on whether the inequalit was looking for positive or negative solutions. Eamples: Solve the following inequalities and graph the solution. < 6 ( ) ( ) 5 + > 5 + 9

3 Section B Parametric Equations If f and g are continuous functions on an interval, then the equations = f(t) and = g(t) are called parametric equations and t is the parameter. For t 0, a particle is moving along a curve so that the position of the particle at an time t is given b ( ) = + ( ) t t t t a) Complete the table below: t 0 t = t t+ b) Sketch the path of the particle in the -plane below. Indicate the direction of motion along the path

4 c) At t =, is the particle moving to the left or to the right? Justif our answer. d) At t =, is the particle moving up or down? Justif our answer. e) At what point(s) (, ), if an, is the particle at rest?

5 For t 0, a particle is moving along a curve so that the position of the particle at an time t is given b = + = ( t ) t t a) Complete the table below: t 0 b) Sketch the path of the particle in the -plane below. Indicate the direction of motion along the path

6 c) At t =, is the particle moving to the left or to the right? Justif our answer. d) At t =, is the particle moving up or down? Justif our answer. e) At what point(s) (, ), if an, is the particle at rest?

7 Section C Polar Equations Each point P in the plane can be assigned polar coordinates (r, θ ) where r is the distance from the origin to point P and ϴ is the directed angle to point P measured in radians. To convert back and forth between rectangular coordinates and polar coordinates use the following: (r, θ ) or (, ) r = r cosθ + = r θ = r sinθ tanθ = Eample: Given the polar curve r = sinθ for 0 θ a) Write epressions for and in terms of θ. b) Complete tables below and use them to sketch the graph of the polar curve r = sinθ for 0 θ θ r

8 Eample: Given the polar curve r = cos θ for 0 θ a) Write epressions for and in terms of θ. b) Complete tables below and use them to sketch the graph of the polar curve r = cos θ for 0 θ θ r

9 Eample: Given the graphs of the polar curves r ( θ ) = sin + and r = for 0 θ, find all the angles, θ, where the graphs intersect and label them on the graph. Eample: Given the graphs of the polar curves r ( θ ) = cos 5 + and r = for 0 θ, find all the angles, θ, where the graphs intersect and label them on the graph. 5 5

10 Section D Infinite Geometric Series A series is a sum of a sequence. Eample: an = a+ a + a + a +... ( ) n Partial sums are finite series = n = S a a S = = n = + S a a a S = = n = + + S a a a a S = How to tell if an Infinite Geometric Series converges or diverges n For an infinite geometric series, a( r) a, the series converges if < r < and has a sum S =. r Eamples: Find the first three partial sums of the series. State whether the series converges or diverges. If the series converges, find its sum. n n n Eample: Find the common ratio, r, and, determine the -values for which the series converges, and then find its sum. ( ) + ( ) + ( ) + ( ) ( )

11 AP Calculus BC Summer Review Packet (80 homework points Due Monda 8/0/8, 0 points etra credit if completed and turned in b 8/8/8) This packet is designed to help ou review and build upon some of the important mathematical concepts and skills that ou have learned in our previous mathematics classes that ou will be using in AP Calculus BC. Before ou work on each section, go to m website and watch the corresponding videos and fill out the prerequisite notes for that section. The first eam of the ear will be on Frida 8//8 and will be based on the concepts and methods ou will be practicing in the problems of this packet. SHOW ALL WORK ON PACKET OR SEPARATE SHEETS OF PAPER Section A Solving Absolute Value Inequalities Solve the following inequalities and graph the solution.. + < > 8 8. ( ). 9 + < > 9. ( ) 9 +. < <. + < 8 Section B Parametric Equations. For t 0, a particle is moving along a curve so that the position of the particle at an time t is given b ( ) = ( ) t t t t a) Complete the table below: t 0 t = t 8t b) Sketch the path of the particle in the -plane. Indicate the direction of motion along the path. c) At t =, is the particle moving to the left or to the right? Justif our answer. d) At t =, is the particle moving up or down? Justif our answer. e) At what point(s) (, ), if an, is the particle at rest?

12 . For t 0, a particle is moving along a curve so that the position of the particle at an time t is given b ( ) ( 5 t) t t = + 6t t = ( ) a) Complete the table below: t 0 b) Sketch the path of the particle in the -plane. Indicate the direction of motion along the path. c) At t =, is the particle moving to the left or to the right? Justif our answer. d) At t =, is the particle moving up or down? Justif our answer. e) At what point(s) (, ), if an, is the particle at rest?. For t 0, a particle is moving along a curve so that the position of the particle at an time t is given b = t 9t + t t t = a) Complete the table below: t 0 b) Sketch the path of the particle in the -plane. Indicate the direction of motion along the path. c) At t =, is the particle moving to the left or to the right? Justif our answer. d) At t =, is the particle moving up or down? Justif our answer. e) At what point(s) (, ), if an, is the particle at rest?. For t 0, a particle is moving along a curve so that the position of the particle at an time t is given b = cost = sin t a) Complete the table below: t 0 b) Sketch the path of the particle in the -plane. Indicate the direction of motion along the path. c) At t =, is the particle moving to the left or to the right? Justif our answer. d) At t =, is the particle moving up or down? Justif our answer. e) At what point(s) (, ), if an, is the particle at rest?

13 Section C Polar Equations. Given the polar curve r = + cosθ for 0 θ a) Write epressions for and in terms of θ. b) Complete table below and use them to sketch the graph of the polar curve r = + cosθ for 0 θ θ r

14 . Given the polar curve r = sin θ for 0 θ a) Write epressions for and in terms of θ. b) Complete table below and use them to sketch the graph of the polar curve r = sin θ for 0 θ θ r

15 . Given the graphs of the polar curves r ( θ ) = sin 5 + and r = for 0 θ, find all the angles, θ, where the graphs intersect and label them on the graph Given the graphs of the polar curves r ( θ ) = cos + and r = for 0 θ, find all the angles, θ, where the graphs intersect and label them on the graph Given the graphs of the polar curves r ( θ ) = cos + and r = for 0 θ, find all the angles, θ, where the graphs intersect and label them on the graph

16 Section D Infinite Geometric Series Find the first three partial sums of the series. State whether the series converges or diverges. If the series converges, find its sum.. n. n+. 5 n n+ e n+ n n n n 8. n Find the common ratio, r, and, determine the -values for which the series converges, and then find its sum ( ) ( ) ( ) 0. ( ) ( ) ( ) ( ) ( ) ( ) ( )

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