exam 2 notes Flashcards

1
Q

Chain rule
f(x) = cos(x
^2)

A

-sin(x^2)*2x

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2
Q

Chain rule
g(x) = e^x2+3x−1

A

(2x+3)e^x^2 + 3x-1

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3
Q

Chain rule:h(x) = √
5x^2 − 1

A

1
/ *10
2square5x^2-1

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4
Q

Chain rule
j(x) = sin^3
(4x + 1)

A

12sin^2(4x+1)cos(4x+1)

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5
Q

chain rule:
k(x) = tan(xe^sin x
)

A

sec^2(xe^sinx)[xe^sin x down cos x+e^sinx

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6
Q

Chain rule:
5^x

A

ln(5)*5^x

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7
Q

chain rule:
3^sinx

A

ln(3)*3 ^sin(x) down cos(x)

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8
Q

chain rule:
sec(π^x)

A

sec(pi ^x)tan(pi^x)*(ln pi)pi^x

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9
Q

implicit differentiation:
y^3 + 2x^3 − y = x^2

A

2x-6x^2/3y^2 -1

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10
Q

implicit differentaition:
x + sin(y) = xy

A

y-1/cos(y)x

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11
Q

implicit differentiation:Determine all points where the tangent line to x^4 + y^2 = 3 is horizontal or vertical.

A

Horizontal:(0,+_3
Vertical:(+-4 square 3,0

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12
Q

Implicit differentiation:
Before the development of the Calculus, Fermat had developed a method of finding tangents. When Descartes was informed of Fermat’s method, he didn’t believe it and challenging Fermat to find the
tangent to the curve x^3 + y
^3 = 2xy, predicting that he would fail. Descartes was unable to solve the
problem himself and was intensely irritated when Fermat solved it easily.
-Given x^3 + y^3 = 2xy determine the equation of the tangent line at the point (1, 1)
-A normal line is defined to be the line perpendicular to the tangent line at the point of tangency. Find
the normal line to x^3 + y
^3 = 2xy at the point (1, 1).

A

a.-x+2
b.y=x

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13
Q

Differentiation of Log, Inverse Trig
ln(x^2 + 1) =

A

2x/x^2+1

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14
Q

Differentiation of Log, Inverse Trig
log down 5(x^3 − tan(x)) =

A

3x^2-sec^2(x)/ln(5)(x^3-tanx)

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15
Q

Differentiation of Log, Inverse Trig
ln(x^2sin(x)) =

A

2xsin(x)+z^2cosx/x^2sin(x)

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16
Q

Differentiation of Log, Inverse Trig
squarex^2−1/
x cos(x) =

A

[x/x^2-1 - 1/2x -sinx/2cosx] [square x2-1/xcos(x)

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17
Q

Differentiation of Log, Inverse Trig
x^sin x =

A

[sinx/x + cosxlnx]x^sinx

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18
Q

Differentiation of Log, Inverse Trig
cos^−1 √
x =

A

-1
/
2 square x-x2

19
Q

Differentiation of Log, Inverse Trig
ln(sec^−1 x)

A

1
/
sec-1(x)|x|square x2-1

20
Q

Differentiation of Log, Inverse Trig
sin^−1(tan x) =

A

sec2x
/
square
1-tan2x

21
Q

Differentiation of Log, Inverse Trig
sin(tan^−1 x)

A

cos(tan^-1x)1/x2+1

22
Q

Chain rule:
Air is being pumped into a spherical balloon at a rate of 10 cm3/min. Determine the
rate at which the radius of the balloon is increasing when the radius of the balloon is
5 cm.

23
Q

Chain rule:
A 15 foot ladder is leaning against the wall. The bottom of it is pushed towards the
wall at a rate of 0.5 ft/sec. How fast is the top of the ladder moving up the wall when
the bottom is 9 feet away from the wall?

24
Q

Chain rule:
Suppose the earth is fixed at the origin and a comet is traveling along the parabola
y = 2 − x^2
in such a way that the comet’s x coordinate is always increasing at a rate
of 0.1 units per day. An earth-based telescope is continually pointed at the comet.
When the comet is at the point (1, 1), how fast is the angle of the telescope going to
be changing?

25
related rates: A tank of water in the shape of a cone is leaking water at a constant rate of 2 f t3/hour. The base radius of the tank is 5 f t and the height of the tank is 14 f t. At what rate is the depth of the water in the tank changing when the depth of the water is 6 f t?
196/15pi
26
l hospital: limx→∞ x^2 + 3x − 1/ 3x^2 − 1
1/3
27
l hospital lim x→2 sin(πx)/ x^2 − 4
pi/2
28
l hospital lim x→2 cos(πx)/ x^2 + 4
1/8
29
l hospital lim x→∞ ln x/ √x
0
30
l hospital lim x→0 sin(3x)/ sin(5x)
3/5
31
l hospital lim x→∞ x^2e^x
infinty
32
l hospital limx→-∞ x^2e^x
0
33
l hospital limx→0 sin^−1(2x)/ tan−1(3x)
2/3
34
* 0 · 0 * 0 · ∞ * ∞ · ∞
0 ? infinity
35
∞ + ∞ * 0 − ∞ * ∞ − ∞
infinity -infinity ?
36
−∞ + ∞ * 0/ 0 * ∞/ 0
? ? infinity
37
0/ ∞ * ∞/ ∞ * 0^ 0
0 ? ?
38
∞^∞ * 0^∞ * ∞^0
infinity 0 ?
39
1^0 * 1^∞
1 1
40
l hospital lim x→−∞ xe^x
0
41
l hospital lim x→0+ √x ln(x)
0
42
l hospital: limx→∞ x^1/x
1
43
l hospital: imx→∞ [e^x + x]^1/x
e
44
limx→∞ x(e^1/x − 1)
1