x - MSU math

MTH 132 - Section 17 Exam 2-Solutions 1. (25%) Consider the following function: f (x) = 5x 2 − 3x + 7 a) (15%) Use the limit definition to compute the derivative f '(x) . €
lim
h→0
f (x
+ h) − f (x) = ... = 10x − 3
€
h
b) (10%) Find the linearization L(x) of f (x) at a = 1. €
L(x) = f(a) + f '(a)(x - a)
€
L(x) = 9 + 7(x -1) = 7x + 2
2. (15%) Assume that: €
⎧2 + x, x ≥1
⎪
f (x) = ⎨ 1
5
⎪ x + , x < 1.
⎩ 2
2
Show that f is differentiable at x=1. Present all the calculations and justify your answer. €
d(2 + x ) 1 When x →1+ ,
=
dx
2
1
5
d( x + ) 1
2 =
When x →1- , 2
dx
2
€
3. (30%) Differentiate the following functions: y = x10 − x 3 + x +10 y' = 10x 9 − 3x 2 +1 •
€
€
€
€
y = x 3 tan x y' = x 2 (3tan x + x sec 2 x) •
y = (x 3 + 7x −1)(5x + 2)
•
y' = 20x 3 + 6x 2 + 70x + 9
€
y = 5x 2 + sin x cos x
•
€
•
€
y' = 10x + cos2 x − sin 2 x
x2
y=
x +1
2x
x 2 y' =
−
x +1 (x +1) 2
4 sin x
2x + cos x
4(2x cos x − 2sin x +1) y' =
(2x + cos x) 2
y=
•
€
4. (15%) Use the chain rule to differentiate the following functions: 1
−5 3
y = (4 x + x )
2
1
−5 − 3
y' = (4 x + x ) (4 − 5x −6 )
3
•
€
y = sin(cos(2x + 5))
y' = −2cos(cos(2x + 5))sin(2x + 5) •
y = 3tan( x)
1
y' =
3sec 2 ( x) 2 x
€
•
€
5. (5%) Assume that y is a function of x . Find y' =
dy
x2
= − €2 dx
y
€
dy
for x 3 + y 3 = 4 . dx
€
€
€
€
€
6. (5%) A spherical balloon is inflated so that its radius increases at a rate of 1 cm/min. How fast is the volume increasing when the diameter is 2000 cm? 4
The volume (V) of a sphere is πr 3 . 3
dV = 4000000π €
7. (5%) The radius of a circle is increased from 2.00 to 2.02 m. Estimate the resulting change in area. The area (A) of a circle is πr 2 . dA = 0.08π €

Download Report