Final Practice: 1. Find the inverse function f 1 (x) of f (x) = 2. Use a linear approximation to estimate 3 x2 3x + 4
7.7.
3. Use the Limit Denition of the Derivative to nd the derivative of f (x) = 2x + 3 at the point a = 3. NO CREDIT will be given if L
Practice Problems on Section 11.3 - ANSWERS
1. a)
f
x
= 2x sin y + y 2 sin x,
b)
f
x
=
2
, f
1+(2x+3y )2 y
c)
f
x
= 4x3 ln(6y x)
d)
f
x
= 3 tan2 (4xy 3 ) sec2 (4xy 3 ) 4y 3 ,
e) f =
x
2.
2f
y2
2f
x2
=
f
y
= x2 cos y 2y cos x
3
1+(2x+3y )2
x4
, f
6y x y
Practice Problems on Sections 10.8, 10.9, 11.2
1. A boy throws a rock up at a 30o angle with the horizontal from a
120 m high cli with the speed of 20 m/s.
If the air resistance is negligent, and the acceleration due to gravity is
approximately 10 m/s2 ,
Practice Problems on Sections 10.7, 10.8 - ANSWERS
1.
a) x = 2t + 1, y = 1 t + 2, z = 8t + 2
2
b) < 1 , 8
3
c)
1
3/ 2 6 3/ 2
2
1
2t+6
4t2 +
0
2.
a) T(t) =<
+ 64t6 dt
1
, sin(3t) , cos(3t)
2
2
2
b) T( ) =<
3
c) (s) =<
r
,2 >
5
1
1
, 0,
2
2
>
>
s
s
s
,
Practice Problems on Section 10.6
For each of the equations below:
a) Identify (name) the surface;
b) Indicate the axis along which the surface is extended (if any);
c) Sketch the surface.
1. x = 2z 2 + y 2 ;
2. y 2 + 4z 2 x2 = 1;
3.
x2
4
= y 2 + 9z 2 ;
4
Practice Problems on Sections 10.4, 10.5
1. Given 3 vectors =< 1, 5, 2 >, =< 2, 1, 1 >, and
a
b
=< 3, 3, 0 >
c
a) Find .
ab
b) Find two unit vectors perpendicular to both and .
a
b
c) Determine whether vectors and are parallel.
a
c
d) Determine whether v
Practice Problems on Sections 10.2 and 10.3
1. Given vectors =< 1, 1, 4 > and =< 1, 2, 2 > .
a
b
a) Find 3 2 .
a
b
b) Find .
ab
c) Find the angle between and .
a
b
d) Find two unit vectors parallel to
a
e) Find two vectors of length 5 parallel to
b
2. G
1. (10 pts.) Find all values of the constant c that make the function f (x) continuous
on (, ).
c2 x2 3x 1, if x < 1
f (x) =
3c cos(x 1), if x 1
answer: Continuity at x = 1 must have lim f (x) = lim+ f (x) = f (1). Therefore,
x 1
2
x 1
2
c (1) 3(1) 1 = 3c
1. (10 pts.) Find all values of the constant c that make the function f (x) continuous
on (, ).
c2 x2 3x 1, if x < 1
f (x) =
3c cos(x 1), if x 1
2. (10 pts.) Find the linear approximation of the function f (x) = sin (2x) at a =
.
6
3. (10 pts.) Find the d
Final Practice:
x2
1. Find the inverse function f 1 (x) of f (x) =
3x + 4
y2
Set x =
and multiply both sides by 3y + 4.
3y + 4
f 1 (x) =
4x + 2
3x 1
2. Use a linear approximation to estimate 3 7.7.
1
1
x0 = 8, f (x) = 3 x then f (x) = 2/3 and y = 2 + (x
Practice Problems on Sections 12.1, 12.2
1. Evaluate the integral
ye2xy dA, where
R
R = cfw_(x, y )|0 x 2, 0 y 3
2. Evaluate the integral
xy dA, where D is the region bounded by the
D
curves y = x2 4x + 3 and y = x 1
3. Evaluate the integral
curves y = x2