Math 151 WIR, Spring 2010, c Benjamin Aurispa
Sections 3.2 & 3.4
1. Differentiate the following functions.
(a) f ( x ) = 9 x + 3
√ x +
(b) f ( x ) = (5 x 5 − 7 x 3
5
√
3 x
+ π
2
+ 9 x +
√
5)(3 x 6 − 10 x 2 + e 5 + cos 3)
4 t
4
+ 3 t − 2
(c) g ( t ) =
(d) h ( x ) = t
5 x x 3
2
+ 7 x
−
√
4 9
2. Given that f (2) = 5 and f
0
(2) = 1, find g
0
(2) if g ( x ) = ( x
3
+ 1)( f ( x ) + 5 x ).
3. Consider the function f ( x ) = 2 x ( x
2
+ 1).
(a) Find the values of x for which the tangent line to the graph of f is parallel to the line 8 x − 2 y = 9.
(b) For what values of a and b is the line ax + by = 6 tangent to the graph of f at x = 1?
4. Find the equation of the tangent line to the graph of f ( x ) = x x 2 + 5 at x = 1.
5. At what points on the graph of f ( x ) = − x
2
+ 4 does the tangent line also pass through the point
(1 , 7)?
6. Find f
0
( x ) for the function below. Where is f not differentiable?
 f ( x ) =




4 x + 11 if x ≤ − 2
6 − x
2 if − 2 < x < 2
− 2 x + 6 if x ≥ 2
7. Given f ( x ) below, find the values of a and b that make f differentiable everywhere.
f ( x ) =
( ax + b if x ≤ 3 x
2 − x if x > 3
8. Calculate the following limits.
sin 9 x
(a) lim x → 0 x (cos x + 1)
(b) lim x → 0
(cos x − 1) sin 5 x x 2 cot 3 x
(c) lim x → 0 csc 4 tan
2 x
3 x
(d) lim x → 0 6 x 2
9. Find the derivatives of the following functions.
(a) f ( x ) = sec x cot x + csc x x − cos x
(b) g ( x ) = tan x + sin x
10. Find the tangent line to the graph of f ( x ) = tan x + 4 at x =
π
4
.
11. For what values of x does f ( x ) = sin x − cos x have a horizontal tangent line?
1