Summer Review Packet 8

AP Calculus Review Worksheet 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.

Simplifying Rational Expressions Simplify. (Show your work!) x−4 1. 2 x − 3x − 4 2.

x3 − 8 x−2

3.

5− x x 2 − 25

4.

x 2 − 4 x − 32 x 2 − 16

II.

Trigonometric Identities

1.

Pythagorean Identities

2.

cos2x=_____________ ____________ ____________ sin2x=___________

3. III.

Operations with Rational Expressions 1 1 − 1. x+h x

2.

2 x2 10 x5

__________________ __________________ __________________

3.

1 1 − 3+ x 3 x

4.

2x 1 8 − − 2 x − 6x + 9 x + 1 x − 2x − 3

IV.

2

Solving equations Solve for Z 1. 4x+10yz=0

2. y 2 + 3 yz − 8 z − 4 x = 0

V.

Operations with functions If

f(x)={(3,5), (2,4), (1,7)} g(x)= x − 3 k(x)= x 2 + 5 determine the following: 1. 2. 3. 4. 5. 6. 7. 8.

VI.

h(x)={(3,2), (4,3), (1,6)}

(f+g)(1)= (k-g)(5)= ( f o h)(3) =

( g o k )(7) = f −1 ( x) =

k −1 ( x) = 1 f ( x) (kg)(x)=

Miscellaneous: Follow the directions for each problem.

f ( x + h) − f ( x ) and simplify if f(x)= x 2 − 2 x. h 3 2. Expand ( x + y )

1. Evaluate

3 2

5 2

3. Simplify: x ( x + x − x 2 ) 4. Eliminate the parameter and write a rectangular equation for x = t2 + 3 y = 2t VII.

Series

Expand and simplify. 4 n2 1. ∑ n=0 2 2.

3

1

∑n n =1

3

VIII. Simplifying Expressions Simplify. 1.

x x

2. eln 3

3. e(1+ ln x )

4. ln 1

5. ln e7

1 6. log 3 ( ) 3

7. log 1 8

8. ln

2

10.

4 xy −2 −

1 3

12 x y 5 3 3 2

13. (4a )

IX.

1 2

11. 27

2 3

9. e3ln x 2 3

3 2

12. (5a )(4a )

−5

14.

3(n + 1)! 5n !

Using the point-slope form y − y1 = m( x − x1 ) , write an equation for the line 1. with a slope of -2, containing the point (3,4)

2. 3. 4. 5.

X.

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

Trigonometry Without a calculator, determine the exact value of each expression. 1. sin 0

5.

cos

9. tan

2.

7π 6

2π 3

12. Sin −1 (sin XI.

6. cos

sin

3. sin

2

π

3π 4

7. tan

3

10. tan

7π ) 6

π

π

7π 4

4. cos π

8. tan

π 6

1 11. cos( S in −1 ) 2

2

Domain and Range For each function, determine its domain and range. 1.

y = x−4

2. y = x 2 − 4 3. y = 4 − x 2 4. y = x 2 + 4 XII. Determine all points of intersection

y = x 2 + 3x − 4 1. y = 5 x + 11

y = cos x 2.

y = sin x in the 1st quadrant

XIII. Solving equations Solve for x, where x is a real number. Show your work. 1.

x 2 + 3x − 4 = 14 x4 −1 =0 2. x3 3. ( x − 5) 2 = 9

4. 2 x 2 + 5 x = 8 5. ( x + 3)( x − 3) > 0 6. x 2 − 2 x − 15 ≤ 0 7. 12 x 2 = 3x 8. sin 2 x = sin x , 0 ≤ x ≤ 2π 9. x − 3 < 7 10. ( x + 1) 2 ( x − 2) + ( x + 1)( x − 2) 2 = 0 11. 27 2 x = 9 x −3 12. log x + log( x − 3) = 1 13. e3 x = 5 14. ln y = 2 x − 3