Additional Review for Test 2

MTH-140 Extra Review for Test 2
Find an equation of the tangent line at x = a.
1) y = x 3 - 9x + 2; a = 3
2) Find an equation for the tangent to the curve y =
10x
at the point (1, 5).
x2 + 1
3) The curves y = ax2 + b and y = 2x2 + cx have a common tangent line at the point (-1, 0). Find a, b, and c.
4) Find all points (x, y) on the graph of y =
x
with tangent lines perpendicular to the line y = 5x - 5.
(x - 5)
Find the second derivative.
5) y = 3x3 - 7x2 + 3ex
Suppose u and v are differentiable functions of x. Use the given values of the functions and their derivatives to find the
value of the indicated derivative.
6) u(1) = 3, u (1) = -7, v(1) = 7, v (1) = -3.
d v
at x = 1
dx u
7) u(2) = 10, u (2) = 4, v(2) = -2, v (2) = -4.
d
(2u - 4v) at x = 2
dx
Find y .
8) y = x +
9) y =
1
x
x-
1
x
x2 + 8x + 3
x
10) y =
(x + 3)(x + 1)
(x - 3)(x - 1)
11) y =
3et
2et + 1
12) y = 6x2 e-x
13) y= t5 tan t -
t
14) y = (csc x + cot x)(csc x - cot x)
1
15) y =
sin x
8x
+
8x
sin x
16) Does the graph of the function y = tan x - x have any horizontal tangents in the interval 0 x
2 ? If so, where?
17) Does the graph of the function y = 12x + 6 sin x have any horizontal tangents in the interval 0 x 2 ? If so,
where?
Suppose that the functions f and g and their derivatives with respect to x have the following values at the given values of
x. Find the derivative with respect to x of the given combination at the given value of x.
x f(x) g(x) f (x) g (x)
18) 3 1
4
6
7
-4
4 -3
3
5
g(x), x = 3
x f(x) g(x) f (x) g (x)
19) 3 1 16
6
7
-6
4 3
3
5
f2 (x) · g(x), x = 3
Find the derivative of the function.
cos x 6
20) h(x) =
1 + sin x
Find dy/dt.
21) y = cos 4 ( t - 17)
22) y = ecos(t/5)
4
Find the derivative of y with respect to x, t, or , as appropriate.
1+ x
23) y = ln
x5
24) y = ln(cos(ln ))
Find the derivative of the function.
5/2
25) y = log 9 (3x2 - 2x) |
Use logarithmic differentiation to find the derivative of y with respect to the independent variable.
26) y = (x + 10)x
27) y = (x + 2) sin x
2
Find the derivative of y with respect to x.
8x
28) y = -cot-1
3
29) y = -sin-1 (3x 2 - 4)
30) y = tan-1
5x
31) y = tan-1 (ln 4x)
4x + 15
32) y = -csc-1
3
3
Answer Key
Testname: MTH-140 EXTRA REVIEW TEST 2
1)
2)
3)
4)
y = 18x - 52
y=5
a = 1, b = -1, c = 2
(0, 0), (10, 2)
5) 18x - 14 + 3ex
40
6)
9
7) 24
8) 2x +
2
x3
9) y =
3x2 + 8x - 3
2x3/2
10) y =
11)
-8x2 + 24
(x - 3)2 (x - 1)2
3et
(2et + 1)2
12) 6xe-x(2 - x)
ds
1
= t5 sec2 t + 5t4 tan t 13)
dt
2 t
14) y = 0
dy x cos x - sin x 8 sin x - 8x cos x
=
+
15)
dx
8x2
sin 2 x
16) Yes, at x = 0, x = , x = 2
17) No
7
18)
4
19) 199
20)
-6 cos5 x
(1 + sin x)6
21) -4 cos3 ( t - 17) sin( t - 17)
4
t 4 cos(t/5)
e
22) - sin
5
5
23)
-10 - 9 x
2x(1 + x )
24) 25)
tan(ln )
5(3x - 1)
ln 9(3x2 - 2x)
26) (x + 10)x ln(x + 10) +
x
x + 10
27) (x + 2) sin x cos x ln (x + 2) +
sin x
x+2
4
Answer Key
Testname: MTH-140 EXTRA REVIEW TEST 2
28)
29)
30)
31)
32)
24
64x2 + 9
-6x
1 - (3x2 - 4)2
5
2(1 + 5x) 5x
1
x(1 + (ln 4x)2 )
12
(4x + 15) (4x + 15)2 - 9
5

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