Summer Assignment

MAT 520: AP Calculus BC

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Knowledge for the Journey

Summer Math Review Packet MAT 520: AP Calculus BC The following packet contains topics that you will be required to know in order to succeed in A.P. Calculus BC this year. You are advised to be familiar with each of the concepts and to turn in the included problems on the first day of school. All of these topics were discussed in either Algebra II or Pre-calculus and will be used frequently throughout the year. Due on the first day of school. Copy each problem onto a separate paper and solve. Show your procedure, not just your answer. The circled answers should be written in this paper in he space provided! All graphs must be shown on this paper! There will be a test covering this materials.

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1. Expand the following binomials a. Expand (x − 2h)5 b. Expand 2x2 − 3( )4 2. Evaluate:

a. ( )ln232ln eee

⎛ ⎞⋅ ⎜ ⎟

⎝ ⎠ b.

1k− 1k +1

⎛⎝⎜

⎞⎠⎟k=3

31

∑ c.

1k− 1

k +1⎛⎝⎜

⎞⎠⎟k=3

∞

∑

3. Write each of the following expressions in the form qpbca where c, p and q are numbers: Find all real solutions to the following without using a calculator.

a. x 12 − 64x 6 = 0 b. 4x 6 − 16x 4 − 25x2 + 100 = 0 c. 8x3 + 27 = 0 d. x 8 − 1 = 0 4. Expand the summation notation and simplify the final answer.

a. 24

0 2n

n=∑

b. 3

31

1n n=∑

5. Solve the following equations for the indicated variables:

(a) x

a+ yb+ zc= 1, for a

(b) 2x4π

+ 1− x2

= 0 , for x

(c) rpositiveforrhrA ,22 2 ππ +=

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6. For the following equations, complete the square and reduce to one of the standard forms

22 )()( byAaxoraxAby −=−−=− . Then sketch the graph, indicating the vertex, x and y intercepts, and axes of symmetry (if any).

a. 342 ++= xxy b. 0233 2 =++ yxx c. 0969 2 =−−− xyy In Exercise 7, (a) draw the graph of the function labeling key points. 7b)D: Then find its (b) domain and (c) range.

7.

24 ,( ) (3/ 2) 3/ 2,

3,

xf x x

x

⎧ −⎪= +⎨⎪ +⎩

1

1 33

xx

x

<≤ ≤>

7c)R:

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8. Consider a function3

2 1xyx

=−

. Please, answer the following questions:

(a) Is this function odd/even/neither? Prove your answer algebraically. (b) Write the equations of the asymptotes: ____________________________________________ (c) Graph the function (only on this paper!) 7. Factor completely:

a. x 12 − 64x6 b. 4x6 − 16x4 − 25x2 + 100 c. 8x3 + 27 d. x8 − 1 9. Solve for x without using a calculator, then check with a calculator:

a. 3 sin2 x = cos2 x ; 0 ≤ x < 4π b. cos

2 x − sin2 x = sinx ; − 2π < x ≤ 2π

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10. Without using a calculator, evaluate the following.

a. cos(210 )° b. 5sin4π⎛ ⎞

⎜ ⎟⎝ ⎠

c. 1tan ( 1)− − d. 1sin ( 1)− −

e. 9cos4π⎛ ⎞

⎜ ⎟⎝ ⎠ f. 1 3sin

2− ⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

g. 7tan6π⎛ ⎞

⎜ ⎟⎝ ⎠ h. 1cos ( 1)− −

11. What is the area of the region enclosed by 2 23 12 6 15x y x y+ + + = ? 12. Solve the following equations or inequalities:

a. 1 − 2

x − 1 ≤ 1x b. 3x− 2x+1 = 4 c. 0< 2x−4( )−1< 12

d. e4( )x ⋅ex2 = e12 e. 3x+ 4( )2 −6 3x+ 4( )+9 = 0 f. 6 1z = (all 6 answers)

g. 4 5 2 6 0x x− ⋅ + = h. x3 − 7x + 6 > 0 i.

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13. Write a piecewise formula for the function below: 14. The equation 0232312 23 =+−− xxx has a solution 2=x . Find all other solutions without using a calculator. 15. Graph the following relations or functions (only on this paper):

a. ( ) ( )2 22 11

4 9x y− −

− = b. x + 4( )264

+y − 2( )29

=1

y 2

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c.

f (x) =| x + 1 |, if x ≤ 0x − 1x − 2 , if 0 < x ≤ 3log2(x + 1), if x > 3

⎧

⎨⎪⎪

⎩⎪⎪

d. f(x) = |x – 2| - | x + 1|

16. (a) Find the equation of the line which is the perpendicular bisector of the line joining (1,-2) and (6, 10). (b) Find the center and vertices of the ellipse whose equation is (c) If two points on a circle are (3,2) and (6,-1), and an equation of a line that goes through the center of

this circle is y = 2x – 7, find an equation for the circle. (d) Find the coordinates of the points at which the parabola y = x2 intersects the circle with center at the

origin and radius equal to 1. (e) An ellipse which centered at the origin has vertices (4,0), (0,-2),(-4,0), and (0,2). Find its equation. (f) A hyperbola which is centered at the origin intersects the y axis at (0,2) and (0,-2). A point on the

hyperbola has coordinates ( 52,3) . Find an equation for the hyperbola.

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17. Find the domain of each of the following functions. Find the range only for (c) and (d). You may use the calculator where appropriate.

(a) ( )2

3 12

xf xx x

+=+ −

(b) f (x) =

x + 1( ) x − 2( ) x − 3( )1 − x2

(c) g x( ) = 5x2 −5

4x2 −1 (d) ( ) x

f xx

=

(e) f (x) = x2 − 5x + 4

x2 − 4x (f) g(x) = log 2x − 3

x2 − 3x + 2⎛

⎝⎜⎞

⎠⎟

18. Simplify ( ) ( )f x h f xh

+ −, where

a. f (x) = 2x2 + 3x −1 b. f (x) = 3x −1

19. Find the inverses of the functions:

Sketch only in this paper, please.

20. A function ( )y f x= has the graph to the right. Sketch the graph of the inverse function

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21. Express x in terms of the other variables in the picture. (a) (b)

22. Graph the following functions in the polar grids (use the symmetry, please): (a) r = -2 + 3cosθ (b) r = 3sin(2θ )

23. Write the partial fraction decomposition:

(a)

x + 3(x − 2)(x − 1)(x + 1) (b)

x(x2 + 9)(x +1)

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The functions in #24 and #25, are given parametrically.

(a) Eliminate the parameter and represent the functions in terms of x and y only. (b) Graph the functions in this paper only!

24. x = 2cos(2t), y = -3sin(2t) 0 ≤ t ≤ π 25. x = 2t; y = t2 – t – 1 -2 < t < 3

26. Evaluate the following expressions:

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For the Multiple Choice, please provide in this paper, the sentence or algebraic explanation of the chosen answer: 27. 28. 29. 30. 31.

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32. 33. 34. 35. 36.

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37. 38. 39. 40. 41.

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42. 43. 44. 45. You should know the following trigonometric identities. (A) sin(-x) = -sin x (C) cos(x + y) = cos x cos y - sin x sin y (B) cos(-x) = cos x (D) sin(x + y) = sin x cos y + cos x sin y (E) sin2 x + cos2 x=1 (F) sin(π – x) = sinx (G) sin(π + x) = - sinx (H) cos(π – x) = - cosx

(I) cos(π + x) = - cosx (J) sin(π2 − x) = cosx

(K) sin(π2 + x ) = cosx (L) cos(π2 + x ) = − sinx

(M) cos(π2 − x ) = sinx

Use these to derive the following important identities, which you should also know.

a. tan2 x + 1 = sec2 x b. sin(2x) = 2 sin(x) cos(x) c. cos(2x) = cos2 x − sin2 x

d. cos(2x) =2cos2 x − 1 e. cos(2x) = 1 − 2 sin2 x

GOOD LUCK STUDYING!