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calc > exam 2 notes > Flashcards

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.

A

1/10

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?

A

3/8

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?

A

-1

25
Q

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?

A

196/15pi

26
Q

l hospital:
limx→∞
x^2 + 3x − 1/
3x^2 − 1

A

1/3

27
Q

l hospital
lim x→2
sin(πx)/
x^2 − 4

A

pi/2

28
Q

l hospital
lim x→2
cos(πx)/
x^2 + 4

A

1/8

29
Q

l hospital
lim x→∞
ln x/
√x

A

0

30
Q

l hospital
lim x→0
sin(3x)/
sin(5x)

A

3/5

31
Q

l hospital
lim x→∞
x^2e^x

A

infinty

32
Q

l hospital
limx→-∞
x^2e^x

A

0

33
Q

l hospital
limx→0
sin^−1(2x)/
tan−1(3x)

A

2/3

34
Q
  • 0 · 0
  • 0 · ∞
  • ∞ · ∞
A

0
?
infinity

35
Q

∞ + ∞
* 0 − ∞
* ∞ − ∞

A

infinity
-infinity
?

36
Q

−∞ + ∞
*
0/
0
* ∞/
0

A

?
?
infinity

37
Q

0/
∞
* ∞/
∞
* 0^
0

A

0
?
?

38
Q

∞^∞
* 0^∞
* ∞^0

A

infinity
0
?

39
Q

1^0
* 1^∞

A

1
1

40
Q

l hospital
lim x→−∞
xe^x

A

0

41
Q

l hospital
lim
x→0+
√x ln(x)

A

0

42
Q

l hospital:
limx→∞
x^1/x

A

1

43
Q

l hospital:
imx→∞
[e^x + x]^1/x

A

e

44
Q

limx→∞
x(e^1/x − 1)

A

1

calc (1 decks)

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