Fiesta 3 Ab | Solutions | Fall 2009 | Ap Calculus| Www.ilearnmath.net

iLearnMath.net | AP Calculus | Fiesta 3 AB | Solutions  2009   

1. Determine whether the function is increasing, decreasing, or neither. a) f ( x) = 3− x Decreasing. b) f ( x) = x 1+1 2

Neither. c) g (t ) = t 2 + t Neither. d) g (t ) = t 3 + t Increasing. 2. Determine whether the function is even, odd, or neither. Write a short proof of your claim in each case. a) f ( x) = x 4 − 3x 2 Even: f (− x) = (− x) 4 − 3(− x) 2 = x 4 − 3 x 2 = f ( x)

b) f ( x) = sin ( x + 1) Neither. Symmetry about the origin is lost after horizontal shift by 1 unit. c) f ( x) = 2− x Even:

2

f ( x ) = 2− x = 2− ( − x ) = f ( − x) 2

2

3. Find the roots of f ( x) = x 4 − 4 x 2 and sketch its graph. Discuss the increasing/decreasing behavior of this function. f ( x) = x 4 − 4 x 2 = x 2 ( x 2 − 4) = x 2 ( x − 2)( x + 2) x x2 ( x − 2)

( x + 2) f ( x)

−∞

−2 + −

− +

+ −

0 0

+ −

0 0

0

∞

2 + −

+ −

0 0

+ +

+ +

iLearnMath.net | AP Calculus | Fiesta 3 AB | Solutions  2009   

I/D behavior can be described with the aid of a graphing calculator.

4. Let f ( x ) = cos x and g ( x) = x −1 .Compute the composite functions f D g and g D f , and find their domains. f D g = cos( g ( x )) = cos( 1x ) ⇒ D = {x ∈ \ | x ≠ 0} gD f =

1 f ( x)

=

1 cos x

⇒ D = {x ∈ \ | x ≠ π2 + kπ }

5. Let f ( x ) = cos x . Sketch the graph of y = 2 f

 

( 13 x − π4 ) for 0 ≤ x ≤ 6π .