Download MATH2414 EXAM 1 REVIEW Find the derivative of y with respect to

MATH2414 EXAM 1 REVIEW
Find the derivative of y with respect to x, t, or , as appropriate.
ln x
1) y =
x7
Find the derivative of y with respect to the independent variable.
2) y = 8 t
A)
1
2 t
8 t
B) 8 t ln 8
C)
ln 8 t
2 t
D)
Find the derivative of the function.
5/2
3) y = log8 (3x2 - 2x) |
Use logarithmic differentiation to find the derivative of y.
x x5 + 2
4) y =
(x + 9)1/3
5) y =
3 (4x + 1)(x + 2)2
(x3 + 6)(x + 8)
Use logarithmic differentiation to find the derivative of y with respect to the independent variable.
6) y = (x + 8) sin x
7) y = 5xx
2
Find the value of df-1 /dx at x = f(a).
8) f(x) = x3 - 12x2 - 6, x 8, a = 4
9) f(x) = x2 - 4x + 7; a = 5
Find the derivative of y with respect to x.
10x + 13
10) y = sin-1
7
11) y = tan-1 (ln 2x)
12) y = tan-1
5x
Evaluate the integral.
13)
et cot(et - 9 )dt
1
ln 8
8 t
2 t
dx
14)
81x2 - 3
x
ex dx
1 - e2x
15)
dx
16)
xln x5
Use the substitution formula to evaluate the integral.
3 /4
9 + cot csc 2 d
17)
/4
ln
18)
3/5 5 e5x dx
1 + e10x
0
- 2/3
19)
-2/3
4
20)
6-
21)
x
x
1
3 /2
dt
t 9t2 - 1
dx
sin d
2 + cos
2
Find the area of the shaded region.
22)
f(x) = x3 + x2 - 6x
g(x) = 6x
23)
y = 2x2 + x - 6
y = x2 - 4
Find the area enclosed by the given curves.
1
24) y = x2 , y = -x2 + 6
2
25) Find the area of the region in the first quadrant bounded on the left by the y-axis, below by the line y =
above left by y = x + 4, and above right by y = - x2 + 10.
3
1
x,
3
Find the volume of the solid generated by revolving the shaded region about the given axis.
26) About the y-axis
y=
3x
27) About the x-axis
y = 2 sin x
Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.
4
28) y = , y = - x + 5
x
Find the volume of the solid generated by revolving the region about the y-axis.
3
29) The region enclosed by x = , x = 0, y = 1, y = 5
y
Find the volume of the solid generated by revolving the region about the given line.
30) The region in the second quadrant bounded above by the curve y = 4 - x2 , below by the x-axis, and on the right
by the y-axis, about the line x = 1
4
Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated axis.
31) About the x-axis
y=5
x=5
x=
25 - y2
32) About the y-axis
y = 4x - x2
Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and
lines about the x-axis.
33) x = 8 - y2 , x = y2 , y = 0
Find the volume of the solid generated by revolving the region about the given axis. Use the shell or washer method.
34) The region bounded by y = 3 x, y = 3, and x = 0 about the line x = 1
5
Answer Key
Testname: E1 R
1)
1 - 7ln x
x8
2) D
3)
5(3x - 1)
ln 8(3x2 - 2x)
4)
x x5 + 2 1
5x4
1
+
1/3
x
5
3x
+ 27
(x + 9)
2x + 4
5)
1 3 (4x + 1)(x + 2)2
4
2
3x2
1
+
3
3
4x
1
x
2
3
x
+
+
+8
(x + 6)(x + 8)
x +6
6) (x + 8) sin x cos x ln (x + 8) +
sin x
x+8
2
7) 5xx + 1(2 ln 5x + 1)
1
8) 48
9)
1
6
10
10)
11)
12)
49 - (10x + 13)2
1
x(1 + (ln 2x)2 )
5
2(1 + 5x) 5x
13) ln sin(et - 9) + C
3
sec-1 3 3 x + C
14)
3
15) sin-1 (ex) + C
1
16) ln ln x5 + C
5
17) 18
18)
12
19) -
12
20) 9
21) -ln 2
937
22)
12
23)
19
3
24) 16
73
25)
6
6
Answer Key
Testname: E1 R
26)
27
5
27) 2 2 - 4
28) 9
36
29)
5
30)
56
3
31)
125
3
32)
128
3
33) 16
7
34)
5
7