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If you do not have a graphing calculator, please obtain one of the following recommended calculators: TI 84+, TI 89, TI-nspire (CAS or not)*. If you already have a calculator, make sure to update the OS. * CAS is not permitted for the ACT

AP CALCULUS AB

SUMMER ASSIGNMENT 2015

Mrs. Brander

I. Basic Algebraic Rules 1. Are the following statements true? If not, change them to make them true. a.

2k k = 2k + 4 k + 4

b.

a 3a = b 3b

! " #$ !%$

c.

a 3a = b b

d. 3

g.

1 1 1 = + p+q p q

x+ y x y = + 2 2 2

a + b 3a + b = c c

e. 3

f. 3

='+1

'

II. Complex Fractions & Rational Expressions 2. Simplify.

x a. 2 x 4

b. h ÷

x−2 +

x+h h

c.

5 x−2

x−2

3. Write as a single fraction with the denominator in factored form. a.

3 2 − x +1 x

7 x2 + 5x 5x − 2 2 x +1 x −6

c. x (1 − 2 x )

−

3 2

+ (1 − 2 x )

b. 2 0

−

1 2

d.

( 3x − 2 )

1 2

+ x ( 3x − 2 )

−

1 2

2 −3 x e. 1 1− x −1

4. Evaluate without a calculator: a. *64-// 0 b. *161/2 0

c. *27-/1 0

d. *326/- 0

III. Negative and Fractional Exponents 5. Simplify using only positive exponents. Do not rationalize the denominators.

4 x − 16

a. 4

( x − 4)3

1 4 1 b. −2 + −1 −1 + −2 x y y x

−

1 2

x −2 c. −1 − x y

−3

IV. Solving Equations and Factoring 6. Solve for y’ in simplest form. a. xy’ + y = 1 + y’

b. 3y2y’ + 2yy’ = 5y’ + 2x

c. 3x2yy’ + 2xy2 = 2yy’

7. Solve the quadratic equation. Use any means from algebra: factoring, quadratic formula, graphing. Be sure answers are simplified. If you use graphing, state what you did on the calculator. a. 4 x 2 −21x − 18 = 0

b. 2 x 2 − 3 x + 3 = 0

c. x 4 − 9 x 2 + 8 = 0

8. Factor completely (There should be no fractional or negative exponents.) a. 3 x 3 +192

b. 9 x 2 − 3 x − 2

3

c. 2 x − 6 x 2

d. sin x + tan x

e. e − x − xe − x + 2 x 2 e − x

f. 2x4 + 5x3 - 3x2

V. Equations of lines 9. Find the equation of the line that passes through the point (2, 4) and is parallel to the line 2x + 3y – 8 = 0.

10. Find the equation of the line that is perpendicular to the line 2x + 3y – 8 = 0 at the point (1, 2).

11. The line with slope 5 that passes through the point (-1, 3) intersects the x-axis at a point. What are the coordinates of this point?

12. What are the coordinates of the point at which the line passing through the points (1, -3) and (-2, 4) intersects the y-axis?

13. Graph the equation y = x 3 − x and answer the following questions. a. Is the point (3, 2) on the graph?

b. Is the point (2, 6) on the graph?

c. Is the function odd, even or neither?

d. Find the x and y – intercept(s).

VI. Asymptotes and Intercepts 14. Find all intercepts and asymptotes. (NO slant asymptotes.) a. y = x − 4 x

x 2 + 3x b. y = (3x + 1) 2

x2 − 4 c. y = 2 x − x − 12

d. y =

2

2

3x − 1 2x 2 + x − 6

VII. Domain 15. Use interval notation to identify the domain for each of the following functions. a. h( x ) =

c.

3

1 4 x − 21x − 18 2

x−6

2

b. k ( x ) =

x 2 − 5 x − 14

d. d ( x ) = ln(2 x − 12)

x − x − 30

VIII. Graphing Functions 16. Graph the following functions.

1 x ≤ 0 a. f(x) = −1 x > 0

2 x (−∞, −1) 2 b. f(x) = 2 x [−1, 2) − x + 3 [2, ∞ )

c. f(x) =

16 − x 2

3 − x x ≤ 1 d. y = 2 x x > 1

2 4 − x 3 3 e. y = x + 2 2 x + 2

1 ≤ x ≤ 3 x>3 x