exam 2 notes Flashcards
Chain rule
f(x) = cos(x
^2)
-sin(x^2)*2x
Chain rule
g(x) = e^x2+3x−1
(2x+3)e^x^2 + 3x-1
Chain rule:h(x) = √
5x^2 − 1
1
/ *10
2square5x^2-1
Chain rule
j(x) = sin^3
(4x + 1)
12sin^2(4x+1)cos(4x+1)
chain rule:
k(x) = tan(xe^sin x
)
sec^2(xe^sinx)[xe^sin x down cos x+e^sinx
Chain rule:
5^x
ln(5)*5^x
chain rule:
3^sinx
ln(3)*3 ^sin(x) down cos(x)
chain rule:
sec(π^x)
sec(pi ^x)tan(pi^x)*(ln pi)pi^x
implicit differentiation:
y^3 + 2x^3 − y = x^2
2x-6x^2/3y^2 -1
implicit differentaition:
x + sin(y) = xy
y-1/cos(y)x
implicit differentiation:Determine all points where the tangent line to x^4 + y^2 = 3 is horizontal or vertical.
Horizontal:(0,+_3
Vertical:(+-4 square 3,0
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.-x+2
b.y=x
Differentiation of Log, Inverse Trig
ln(x^2 + 1) =
2x/x^2+1
Differentiation of Log, Inverse Trig
log down 5(x^3 − tan(x)) =
3x^2-sec^2(x)/ln(5)(x^3-tanx)
Differentiation of Log, Inverse Trig
ln(x^2sin(x)) =
2xsin(x)+z^2cosx/x^2sin(x)
Differentiation of Log, Inverse Trig
squarex^2−1/
x cos(x) =
[x/x^2-1 - 1/2x -sinx/2cosx] [square x2-1/xcos(x)
Differentiation of Log, Inverse Trig
x^sin x =
[sinx/x + cosxlnx]x^sinx
Differentiation of Log, Inverse Trig
cos^−1 √
x =
-1
/
2 square x-x2
Differentiation of Log, Inverse Trig
ln(sec^−1 x)
1
/
sec-1(x)|x|square x2-1
Differentiation of Log, Inverse Trig
sin^−1(tan x) =
sec2x
/
square
1-tan2x
Differentiation of Log, Inverse Trig
sin(tan^−1 x)
cos(tan^-1x)1/x2+1
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.
1/10
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?
3/8
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?
-1
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
l hospital:
limx→∞
x^2 + 3x − 1/
3x^2 − 1
1/3
l hospital
lim x→2
sin(πx)/
x^2 − 4
pi/2
l hospital
lim x→2
cos(πx)/
x^2 + 4
1/8
l hospital
lim x→∞
ln x/
√x
0
l hospital
lim x→0
sin(3x)/
sin(5x)
3/5
l hospital
lim x→∞
x^2e^x
infinty
l hospital
limx→-∞
x^2e^x
0
l hospital
limx→0
sin^−1(2x)/
tan−1(3x)
2/3
- 0 · 0
- 0 · ∞
- ∞ · ∞
0
?
infinity
∞ + ∞
* 0 − ∞
* ∞ − ∞
infinity
-infinity
?
−∞ + ∞
*
0/
0
* ∞/
0
?
?
infinity
0/
∞
* ∞/
∞
* 0^
0
0
?
?
∞^∞
* 0^∞
* ∞^0
infinity
0
?
1^0
* 1^∞
1
1
l hospital
lim x→−∞
xe^x
0
l hospital
lim
x→0+
√x ln(x)
0
l hospital:
limx→∞
x^1/x
1
l hospital:
imx→∞
[e^x + x]^1/x
e
limx→∞
x(e^1/x − 1)
1
