Note 6 (Exam2 Review)
[Pages:15]Math 151 Engineering Calculus I Summer 2020
Note 6 (Exam2 Review)
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(1) Find the derivative. (a) f (x) = ex2 key: f (x) = ex2 ? 2x
(b) f (x) = x sin7(cos(6x)) key: f (x) = sin7(cos(6x)) + x ? 7 sin6(cos(6x)) ? cos(cos(6x)) ? (- sin(6x)) ? 6
(c)
f (x)
=
(x - 1)2 ex2+2x
key:
f (x) =
2(x - 1) ? ex2+2x - (x - 1)2 ? ex2+2x ? (2x + 2) e2x2+4x
Copyright c 2020 Dr. JD Kim
Note 6
Page 1 of 15
Math 151 Engineering Calculus I Summer 2020
(d) f (x) = x sec4(5x) key: f (x) = sec4(5x) + x ? 4 sec3(5x) ? sec(5x) ? tan(5x) ? 5
(e) f (x) = cos(x + e3x) key: f (x) = - sin(x + e3x) ? (1 + 3e3x)
(f) f (x) = (4 - x)2 tan x
key:
f (x) =
2(4 - x) ? (-1) ? tan x - (4 - x)2 ? sec2 x tan2 x
Copyright c 2020 Dr. JD Kim
Note 6
Page 2 of 15
(g) f (x) = ln(sin2 x)
key:
f (x) =
2 sin x ? cos x sin2 x
=
2 cos x sin x
Math 151 Engineering Calculus I Summer 2020
(h) g(x) = ln(xe-2x)
key:
f (x) =
e-2x + x ? e-2x(-2) x ? e-2x
(i) f (x) = log5(1 + cos x)
key:
f (x) =
- sin x (1 + cos x) ln 5
Copyright c 2020 Dr. JD Kim
Note 6
Page 3 of 15
(j) f (x) = arcsin(1/x)
key: f (x) =
1
1 -
(
1 x
)2
?
-1 x2
Math 151 Engineering Calculus I Summer 2020
(k) f (x) = 1 - x2 arcsin x
key:
f (x) =
1 2
(1
-
x2)-1/2
?
(-2x)
?
arcsin
x
+
1
(l) f (x) = arctan(x2 - x)
key:
f (x) =
1 1 + (x2 - x)2
? (2x - 1)
Copyright c 2020 Dr. JD Kim
Note 6
Page 4 of 15
Math 151 Engineering Calculus I Summer 2020
dy (2) Find dx.
(a) x2y3 - 5x3 = sec(4y) + 10y2
key:
dy dx
=
2xy3 - 15x2 4 sec(4y) tan(4y) + 2y ? 10y2 ln 10 - 3x2y2
(b) tan(xy2) + sin y = 6x2 + 8y + 2
key:
dy dx
=
12x - y2 ? sec2(xy2) 2xy ? sec2(xy2) + cos y -
8
Copyright c 2020 Dr. JD Kim
Note 6
Page 5 of 15
(3) If f (x) = cos(ln x2), find f (1). key: 0
Math 151 Engineering Calculus I Summer 2020
Copyright c 2020 Dr. JD Kim
Note 6
Page 6 of 15
Math 151 Engineering Calculus I Summer 2020
(4) Use the logarithmic differentiation to find the derivative of the function.
(a) y = xcos x
key:
dy dx
=
xcos x
-
sin
x
ln
x
+
cos x
x
(b) y = (ln x)cos(x2+3) key: y = (ln x)cos(x2+3)
-2x
sin(x2
+
3)
?
ln(ln
x)
+
cos(x2 + x ln x
3)
Copyright c 2020 Dr. JD Kim
Note 6
Page 7 of 15
Math 151 Engineering Calculus I Summer 2020
(5) Given that h(5) = 3, h(5) = -2, g(5) = -3 and g(5) = 6, find f (5) for each of the following. (a) f (x) = g(x)h(x) key: 24
(b)
f (x)
=
g(x) h(x)
key:
4 3
(c) f (x) = g(h(x)) key: Not enough information
Copyright c 2020 Dr. JD Kim
Note 6
Page 8 of 15
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