solve the following for x.

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

x^2-x = 2

I am not sure what to do now

2) e^-5x * e^4 = e

-5x + 4 = 1

x= 3/5

3) Suppose that the price p (in dollars) and the demand in x(thousands) of a commodity satisfy the demand equation 6p + x + xp =94. How fast is the demand changing at times when x= 4 and p=9 and the price is rising at the rate of $2 per week.

Would I take the derivative and then plug in 4 and 9?

4) solve: ln x = 4.5

e^lnx = e^4.5

x=e^4.5
5) evaluate: integral of x ln 4x dx

1/2 x^2 + 8x^2

6) evaluate the integral from 3 to 1 of x^3 - 3x +1

1/4x^4 + 1/6x^3 then plug in 3 and solve and the same for 1 and subtract

7) A bacteria culture grows exponentially according to the equation p(t)=1000e^.1t

a) what is the original amount of bacteria and what is the growth constant

the amount is 1000 and the constant is .1

b) what is the differential equation satisfied by p(t)

this one I am not sure about

c) what is the population after 5 days.

Just plug in 5 to the equation.

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

I bet you mean
1)e^(x^2-x)=e^2
x^2-x = 2
x^2 - x -2 = 0
(x-2)(x+1) = 0
x = 2 or -1

Are the rest that I did do correct?

3) Suppose that the price p (in dollars) and the demand in x(thousands) of a commodity satisfy the demand equation 6p + x + xp =94. How fast is the demand changing at times when x= 4 and p=9 and the price is rising at the rate of $2 per week.

Would I take the derivative and then plug in 4 and 9?
----------------------------------------
6 dp/dt + dx/dt +x dp/dt+ p dx/dt = 0
plug
6*2 + dx/dt + 4*2 + 9 dx/dt = 0
20 + 10 dx/dt = 0
dx/dt = -2

Is number 3 considered a related rates problem?

5) evaluate: integral of x ln 4x dx

= (x^2/2) ln 4x -x^2/4

1)evaluate: integral of x ln 4x dx

1/2 x^2 + 2x^2 Is this correct?

6) evaluate the integral from 3 to 1 of x^3 - 3x +1

1/4x^4 + 3/2 x^2 Is this the correct antiderivative?

6) evaluate the integral from 3 to 1 of x^3 - 3x +1

x^4/4
- 3 x^2 /2
+ x

thank you

Is number 3 considered a related rates problem?

I suppose so.
I should have been more careful about chain rule to show you what I did.
6p + x + xp =94.

6 dp/dx + dx/dx + x dp/dx + p dx/dx = 0

times dx/dt
6 dp/dx*dx/dt + dx/dx*dx/dt + x dp/dx*dx/dt + p dx/dx*dx/dt = 0

6 dp/dt +dx/dt +x dp/dt + p dx/dt = 0

7) A bacteria culture grows exponentially according to the equation p(t)=1000e^.1t

a) what is the original amount of bacteria and what is the growth constant
----------> 1000 because e^0 = 1, k = .1

the amount is 1000 and the constant is .1
------------> right

b) what is the differential equation satisfied by p(t)

this one I am not sure about
----------> dp/dt = .1 p
then of course
dp/p = .1 dt
ln p = .1 t + C
e^lnp = p = e^(.1t + C)
p = e^C e^(.1t) = c e^(.1t)

c) what is the population after 5 days.

Just plug in 5 to the equation.
--------> right