MATH 1680 Flashcards

1
Q

e^2x

A

()I=

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2
Q

4e

A

(4e)I=0

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3
Q

e^2x-1

A

(e^2x-1)I= 2e^2x-1

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4
Q

xe^x

A

(xe^x)I= (1+x)e^x

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5
Q

(e^x-e^-x)/2

A

((e^x-e^-x)/2)I= 1/2(e^x+e^-x= (e^x+e^-x)/2

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6
Q

e^x/x

A

(e^x/x)I= (x3^x-e^x)/x^2

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7
Q

xe^x-e^x

A

(xe^x-e^x)I= xe^x+e^x-e^x= xe^x

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8
Q

y = e^−5x6

A

-30x^5e^{-5x^6}

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9
Q

f(x) = e^x^2 + 8

A

2xe^{x^2+8}

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10
Q

g(x) = 6e√x

A

3e^sqrt{x}/sqrt{x}

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11
Q

y = x2ex − 2xex + 8ex

A

x^2e^x+6e^x

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12
Q

Find an equation of the tangent line to the graph of the function at the given point.
g(x) = ex5 − 6x, (−1, e5)

A

y=-e^5x

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13
Q

Find an equation of the tangent line to the graph of the function at the given point.
y = (e2x − 3)2, (0, 4)

A

y=-8x+4

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14
Q

Find the second derivative.
f(x) = 7e−x − 9e−7x

A
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15
Q

Find
dy
dx
implicitly.
x2y − ey − 9 = 0

A

-\frac{2xy}{x^2-e^y}

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16
Q

Find dy/dx implicitly.
exy + x2 − y2 = 15

A

-\frac{ye^{xy}+2x}{xe^{xy}-2y}

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17
Q

Find the second derivative.
f(x) = (7 + 4x)e−3x

A

-24e^{-3x}+36xe^{-3x}+63e^{-3x}

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18
Q

Differentiate.
y =
x
ex

A

\frac{1-x}{e^x}

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19
Q

Differentiate.
g(x) = 5ex√x

A

\frac{5e^x+10xe^x}{2\sqrt{x}}

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20
Q

Differentiate.
f(x) =
2 − xex
x + ex

A

\frac{-e^{2x}-x^2e^x-2-2e^x}{\left(x+e^x\right)^2}

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21
Q

Find an equation of the tangent line to the given curve at the specified point.
y =
ex
x
, (1, e)

A

y=e

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22
Q

Find the derivative of the function.
f(z) = ez/(z − 8)

A

e^{\left(\frac{z}{z-8}\right)}\cdot -\frac{8}{\left(z-8\right)^2}

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23
Q

Find the derivative of the function.
f(x) = ln(4 − x6)

A

\frac{1}{4-x^6}\cdot \left(-6x^5\right)

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24
Q

Find the derivative of the function.
y = (ln(x6))2

A

\frac{2\ln \left(x^6\right)^2}{\ln \left(x\right)\cdot x}

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25
Find the derivative of the function. y = ln(x) x2
\frac{1-2\ln \left(x\right)}{x^3}
26
Find the derivative of the function. y = ln √(x + 7/x − 7)
-\frac{7}{x^2-49}
27
Constant Rule
∫k dx = kx + C
28
Sum Rule
d/dx [f(x) + g(x)] = d/dx [f(x)] + d/dx [g(x)]
29
Difference Rule
∫[f(x) - g(x)] dx = ∫f(x) dx - ∫g(x) dx.
30
Simple Power Rule
∫xn dx = xn+1/(n + 1) + C, where n ≠ -1
31
Simple Log Rule
∫logax dx = x logax - x/ln a + C. ∫ln x dx = x ln x - x + C.
32
Simple Exponential Rule
∫ax dx = ax/ln a + C. ∫ex dx = ex + C [Because ln e = 1]
33
Constant Multiple Rule
∫kf(x) dx= k ∫ f(x) dx
34
∫-5dx= -5x+C
-5x+C or prime= -5
35
∫3xdx= 3 (x)^1+1/1+1
3/2 x^2 +c
36
USUB (3-4x^2)3 (-8x) dx
(3-4x2)4/4 + C
37
USUB (x3+1)1/2 (3x2) dx
(x3+1)3/2 / (3/2) + C
38
USUB (x-3)5/2
(x-3)7/2 / 7/2 +C
39
USUB (X4+3x2) (6x) dx
USUB (X4+3x2) (6x) + C
40
Identify u and du dx for the integral du/dx dx. (7 − 2x2)5(−4x) dx
u= 7-2x2 du=-4x
41
Find the indefinite integral. Check your result by differentiating. (Use C for the constant of integration.) (x − 6)3⁄2 dx
2/5(x-6)^5/2
42
Identify u and du/dx for the integral du/dx dx. 4x3 Square root of x4 + 3 dx
u= x4+3 du=4x3
43
Find the indefinite integral. Check your result by differentiating. (Use C for the constant of integration.) 5 − 4x2 cube root of (−8x) dx
\frac{3}{4}\left(5-4x^2\right)^{\left(\frac{4}{3}\right)}+C
44
Find the indefinite integral. Check your result by differentiating. (Use C for the constant of integration.) x(1 − 8x2)5 dx
-1/96(1-8x2)6 +C
45
Find the indefinite integral. Check your result by differentiating. (Use C for the constant of integration.) x2/(4x3 + 7)2 dx
-1/(48x3+84)+C
46
Find the indefinite integral. Check your result by differentiating. (Use C for the constant of integration.) (2x3 + 4x)3(3x2 + 2) dx
2x12+16x10+48x8+64x6+32x4+C
47
d(x^n)/dx =
nx^n-1
48
∫ xn dx=
(xn+1) / (n+1) + C
49
Integral of ax is, ∫ ax dx
ax / ln a + C
50
Integral of 1/x is, ∫ 1/x dx
= ln |x| + C
51
Use a symbolic integration utility to find the indefinite integral. (Use C for the constant of integration.) 1 x6 e1⁄(15x5) dx
3e(1/15x5) + C
52
What is ∂ called?
impartial derivative
53
∂f / ∂x = lim h → 0 [ f(x + h, y) - f(x, y) ] / h. ∂f / ∂y = lim h → 0 [ f(x, y + h) - f(x, y) ] / h.
∂f / ∂y = lim h → 0 [ f(x, y + h) - f(x, y) ] / h.
54
Power Rule If u = [f(x,y)]n then,
the partial derivative of u with respect to x and y defined as; ux = n|f(x,y)|n-1∂f/∂x And uy = n|f(x,y)|n-1∂f/∂y
55
56
57
58
59
60
61
Lim as x approaches-2
3/2
62
Lim as x approaches-2
Infinity
63
Lim as x approaches-1+
0
64
Lim as x approaches-1-
1
65
Lim as x approaches-1
0
66
Lim as x approaches infinity
1
67
0/0
Cancel common factors
68
#/0
Infinity if +/+ or -/-