MATH 1680

Question 1

e^2x

Answer 1

()I=

Question 2

4e

Answer 2

(4e)I=0

Question 3

e^2x-1

Answer 3

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

Question 4

xe^x

Answer 4

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

Question 5

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

Answer 5

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

Question 6

e^x/x

Answer 6

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

Question 7

xe^x-e^x

Answer 7

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

Question 8

y = e^−5x6

Answer 8

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

Question 9

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

Answer 9

2xe^{x^2+8}

Question 10

g(x) = 6e√x

Answer 10

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

Question 11

y = x2ex − 2xex + 8ex

Answer 11

x^2e^x+6e^x

Question 12

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

Answer 12

y=-e^5x

Question 13

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

Answer 13

y=-8x+4

Question 14

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

Question 15

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

Answer 15

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

Question 16

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

Answer 16

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

Question 17

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

Answer 17

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

Question 18

Differentiate.
y =
x
ex

Answer 18

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

Question 19

Differentiate.
g(x) = 5ex√x

Answer 19

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

Question 20

Differentiate.
f(x) =
2 − xex
x + ex

Answer 20

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

Question 21

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

Answer 21

y=e

Question 22

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

Answer 22

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

Question 23

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

Answer 23

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

Question 24

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

Answer 24

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

Question 25

Find the derivative of the function.
y =
ln(x)
x2

Answer 25

\frac{1-2\ln \left(x\right)}{x^3}

Question 26

Find the derivative of the function.
y = ln

√(x + 7/x − 7)

Answer 26

-\frac{7}{x^2-49}

Question 27

Constant Rule

Answer 27

∫k dx = kx + C

Question 28

Sum Rule

Answer 28

d/dx [f(x) + g(x)] = d/dx [f(x)] + d/dx [g(x)]

Question 29

Difference Rule

Answer 29

∫[f(x) - g(x)] dx = ∫f(x) dx - ∫g(x) dx.

Question 30

Simple Power Rule

Answer 30

∫xn dx = xn+1/(n + 1) + C, where n ≠ -1

Question 31

Simple Log Rule

Answer 31

∫logax dx = x logax - x/ln a + C. ∫ln x dx = x ln x - x + C.

Question 32

Simple Exponential Rule

Answer 32

∫ax dx = ax/ln a + C. ∫ex dx = ex + C [Because ln e = 1]

Question 33

Constant Multiple Rule

Answer 33

∫kf(x) dx= k ∫ f(x) dx

Question 34

∫-5dx= -5x+C

Answer 34

-5x+C or prime= -5

Question 35

∫3xdx= 3 (x)^1+1/1+1

Answer 35

3/2 x^2 +c

Question 36

USUB (3-4x^2)3 (-8x) dx

Answer 36

(3-4x2)4/4 + C

Question 37

USUB (x3+1)1/2 (3x2) dx

Answer 37

(x3+1)3/2 / (3/2) + C

Question 38

USUB (x-3)5/2

Answer 38

(x-3)7/2 / 7/2 +C

Question 39

USUB (X4+3x2) (6x) dx

Answer 39

USUB (X4+3x2) (6x) + C

Question 40

Identify u and
du
dx
for the integral

du/dx dx.
(7 − 2x2)5(−4x) dx

Answer 40

u= 7-2x2
du=-4x

Question 41

Find the indefinite integral. Check your result by differentiating. (Use C for the constant of integration.)

(x − 6)3⁄2 dx

Answer 41

2/5(x-6)^5/2

Question 42

Identify u and
du/dx for the integral

du/dx dx.
4x3 Square root of x4 + 3 dx

Answer 42

u= x4+3
du=4x3

Question 43

Find the indefinite integral. Check your result by differentiating. (Use C for the constant of integration.)

5 − 4x2 cube root of (−8x) dx

Answer 43

\frac{3}{4}\left(5-4x^2\right)^{\left(\frac{4}{3}\right)}+C

Question 44

Find the indefinite integral. Check your result by differentiating. (Use C for the constant of integration.)

x(1 − 8x2)5 dx

Answer 44

-1/96(1-8x2)6 +C

Question 45

Find the indefinite integral. Check your result by differentiating. (Use C for the constant of integration.)
x2/(4x3 + 7)2 dx

Answer 45

-1/(48x3+84)+C

Question 46

Find the indefinite integral. Check your result by differentiating. (Use C for the constant of integration.)
(2x3 + 4x)3(3x2 + 2) dx

Answer 46

2x12+16x10+48x8+64x6+32x4+C

Question 47

d(x^n)/dx =

Answer 47

nx^n-1

Question 48

∫ xn dx=

Answer 48

(xn+1) / (n+1) + C

Question 49

Integral of ax is, ∫ ax dx

Answer 49

ax / ln a + C

Question 50

Integral of 1/x is, ∫ 1/x dx

Answer 50

= ln |x| + C

Question 51

Use a symbolic integration utility to find the indefinite integral. (Use C for the constant of integration.)
1
x6
e1⁄(15x5) dx

Answer 51

3e(1/15x5) + C

Question 52

What is ∂ called?

Answer 52

impartial derivative

Question 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.

Answer 53

∂f / ∂y = lim h → 0 [ f(x, y + h) - f(x, y) ] / h.

Question 54

Power Rule
If u = [f(x,y)]n then,

Answer 54

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

Question 61

Lim as x approaches-2

Answer 61

3/2

Question 62

Lim as x approaches-2

Answer 62

Infinity

Question 63

Lim as x approaches-1+

Answer 63

0

Question 64

Lim as x approaches-1-

Answer 64

1

Question 65

Lim as x approaches-1

Answer 65

0

Question 66

Lim as x approaches infinity

Answer 66

1

Question 67

0/0

Answer 67

Cancel common factors

Question 68

/0

Answer 68

Infinity if +/+ or -/-