AP Calculus Summer Review Packet

This packet is a review of the entering objectives for AP Calculus and is due on the first day back to school. It is to be done NEATLY and on a SEPARATE sheet of paper. Have a great summer.

I. Simplify. Show the work that leads to your answer. x  4 x3 8 5  x x2 4 x  32 1. 2. 3. 4. x 2  3x  4 x  2 x 2  25 x2 16

II. Trigonometric Identities.

1. Pythagorean = ______2. cos 2x = ______3. sin 2x = ______

III. Simplify each expression. 2 1 1 1 1 x2  2x 1 8 1.  2. 3. 3 x 3 4.   x h x 10 2 2 x x6 x  9 x  1 x  2 x  3 x5

IV. Solve for z: 1. 4x + 10yz = 0 2. y2 + 3yz – 8z – 4x = 0

V. If; f(x) = {(3,5), (2,4), (1,7)} g(x) = x  3 h(x)= {(3,2), (4,3), (1,6)} k(x) = x2 + 5 determine each of the following: 1. (f + h)(1) = 2. (k – g)(5) = 3. (f ◦ h)(3) = 4. (g ◦ k)(7) =

1 5. f -1(x) = 6. k-1(x) = 7. = 8. (kg)(x) = f( x )

VI. Miscellaneous: Follow the directions for each problem.

f( x h )  f ( x ) 1. Evaluate and simplify if f(x) = x2 – 2x. h 2. Expand (x + y)3 3 5 3. Simplify: x2( x x 2  x2 ) 4. Eliminate the parameter and write a rectangular equation for x = t2 + 3 y = 2t

VII. Expand and simplify 4 n2 3 1 1.  2.  3 n0 2 n1 n

VIII. Simplify x 1. 2. eln3 3. e(1 lnx ) x

7 4. ln 1 5. ln e 6. log3(1/3)

1 3ln x 7. log 1/2 8 8. ln 9. 2 e 4xy2 10. 1 11. 272/3 12. (5a2/3)(4a3/2)  12x3 y5 3(n  1)! 13. (4a5/3) 3/2 14. 5n !

IX. Using the point-slope form y – y1 = m(x – x1), write an equation for the line

1. with slope –2, containing the point (3, 4) 1. ______2. containing the points (1, -3) and (-5, 2) 2. ______3. with slope 0, containing the point (4, 2) 3. ______4. parallel to 2x – 3y = 7 and passes through (5 , 1) 4. ______5. perpendicular to the line in problem #1, containing 5. ______the point (3, 4)

X. Given the vectors v = -2i + 5j and w = 3i + 4j, determine 1 1. v 2. w – v 3. length of w 4. the unit vector for v 2

XI. Without a calculator, determine the exact value of each expression.  3 1. sin 0 2. sin 3. sin 4. cos  2 4 7  7  5. cos 6. cos 7. tan 8. tan 6 3 4 6 2  1 7 9. tan 10. tan 11. cos(Sin-1 ) 12. Sin-1(sin ) 3 2 2 6

XII. For each function, determine its domain and range.

1. y x  4 2. y x2  4 3. y4  x2 4. y x2  4

XIII. Determine all points of intersection. 1. parabola y = x2 + 3x –4 and 2. y = cos x and y = sin x in the line y = 5x + 11 first quadrant

XIV. Solve for x, where x is a real number. Show the work that leads to your solution. x 4  1 1. x2 + 3x – 4 = 14 2.  0 3. (x – 5)2 = 9 x 3 4. 2x2 + 5x = 8 5. (x + 3)(x – 3) > 0 6. x2 – 2x - 15  0 7. 12x2 = 3x 8. sin 2x = sin x , 0  x  2 9. |x – 3| < 7 10. (x + 1)2(x – 2) + (x + 1)(x – 2)2 = 0 11. 272x = 9x-3 12. log x + log(x – 3) = 1 13. e3 k = 5 14. ln y = 2t - 3

XV. Graph each function. Give its domain and range. 1. y = sin x 2. y = cos x 3. y = tan x

x  4 4. y = x3 – 2x2 – 3x 5. y = x2 – 6x + 1 6. y = x  1 x 2  4 7. y = 8. y = ex 9. y = x x  2

10. y = 3 x 11. y = ln x 12. y = |x + 3| - 2

x2 if x < 0 1  13. y  14. y =  x + 2 if 0 x  3 x  4 if x > 3

XVI. Graph each of the following. Then, dentify, by name, each polar graph. Give at least one characteristic of each graph (e.g. radius, location, length of petal, point (other than the pole) on the graph, etc.)

1. r = 2 ______

2. r = 3 sin θ ______

3. r = 1 + sin θ ______

4. r = 2cos 3θ ______

XVII. Sequences and series. Evaluate each of the following.

 n 5 4 50  2  2 1.   2)  (2n  4) 3)  n  n 4)  (2n  4) n1  3  n1 n1 n1

st 5. Find the 131 term in the arithmetic sequence in which t3  7 & t9  61.

6. Find the first two terms of the geometric sequence for which t5  36 2 & r  2 .

XVIII. Growth, decay, and logistics.

1. Write a logistic model with the following characteristics…

Limit to growth = 30 Initial Value = 12 Passes through ( 2 , 20 )

2. Write the equation for the exponential model that passes through….

 243  3  2a) ( -2 , 12 ) and ( 0 , 3 ) 2b)  5,  and 1,   2   2 

3. Fill in the blanks…

Domain: ______ x Fill in the blanks for f (x)  2(5) . Range: ______Continuous ( yes/no) _____ Increasing: ______lim f (x)  ______x

Decreasing: ______lim f (x)  ______x

H.A. ______Upper bound: ______

Lower bound: ______