Note 6 (Exam2 Review)

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

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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

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(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

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(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

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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

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