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Find the derivative, dy/dx, for...

Find the derivative, dy/dx, for the following functions:

a. x^2y^4= ln x + 3y

b. y= ln(2x^2+1)

c. y=ln(sqrt x)

d. y= ln x/x

e. y= x^2 ln x

f. y= log2(3x)

2. A popular search engine is targeting European countries where the number

of online households is expected to grow at a steady rate. Data was taken

over time, and it was found that the number of online houses (in millions)

projected can be modeled by the following function:

N(t)= 34.68 + 23.88 ln(1.05t + 5.3), for 0 < t < 2

The values of t are in year and when t= 0, the year is 2004. Address the following questions:

a. What was the projected number of online households at the

beginning of 2005?

b. How fast was the projected number of online households increasing

at the beginning of 2005?

3.. Light is absorbed when it passes through a glass window. If r% of light is

absorbed by a glass with thickness w, then the percent of light that is

absorbed by a piece of glass with a thickness of nw for any natural

number n is modeled by the following function:

A(n)= 100 [1-(1 - r/100)^n], for 0 < r < 100

a. Show that A is an increasing function of n if 0 < r < 100

b. sketch the graeph of A when r = 10

c. Evaluate lim of A(n), as n approaches infinity and interpret the results.

4. Find the derivative for the following:

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

5. The present value of a building in the downtown area is given by the

function

P(t)= 300,000e^-0.09t+sqrt t/2 for 0 < t < 10

Find the optimal present value of the building. (Hint: Use a graphing

utility to graph the function, P(t), and find the value of t0 that gives a point

on the graph, (t0, P(t0)), where the slope of the tangent line is 0.)

6. Find the equation of the line tangent to

f(x)= xe^-x,

at the point where x=0. What does this tell you about the behavior of

the graph when x=0?

7. The unit selling price p (in dollars) and the quantity demanded x (in pairs)

of a certain brand of women’s shoes are given by the demand equation

p(x)= 100e^-0.0001x for 0 < x < 20,000

a.Find the revenue function, R. (Hint: R(x)= x(p(x)), since the

revenue function is the unit selling price at a demand level of x units

times the number of units demanded.)

b. Find the marginal revenue function, R'.

c. What is the marginal revenue when, x=10?

a. x^2y^4= ln x + 3y

b. y= ln(2x^2+1)

c. y=ln(sqrt x)

d. y= ln x/x

e. y= x^2 ln x

f. y= log2(3x)

2. A popular search engine is targeting European countries where the number

of online households is expected to grow at a steady rate. Data was taken

over time, and it was found that the number of online houses (in millions)

projected can be modeled by the following function:

N(t)= 34.68 + 23.88 ln(1.05t + 5.3), for 0 < t < 2

The values of t are in year and when t= 0, the year is 2004. Address the following questions:

a. What was the projected number of online households at the

beginning of 2005?

b. How fast was the projected number of online households increasing

at the beginning of 2005?

3.. Light is absorbed when it passes through a glass window. If r% of light is

absorbed by a glass with thickness w, then the percent of light that is

absorbed by a piece of glass with a thickness of nw for any natural

number n is modeled by the following function:

A(n)= 100 [1-(1 - r/100)^n], for 0 < r < 100

a. Show that A is an increasing function of n if 0 < r < 100

b. sketch the graeph of A when r = 10

c. Evaluate lim of A(n), as n approaches infinity and interpret the results.

4. Find the derivative for the following:

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

5. The present value of a building in the downtown area is given by the

function

P(t)= 300,000e^-0.09t+sqrt t/2 for 0 < t < 10

Find the optimal present value of the building. (Hint: Use a graphing

utility to graph the function, P(t), and find the value of t0 that gives a point

on the graph, (t0, P(t0)), where the slope of the tangent line is 0.)

6. Find the equation of the line tangent to

f(x)= xe^-x,

at the point where x=0. What does this tell you about the behavior of

the graph when x=0?

7. The unit selling price p (in dollars) and the quantity demanded x (in pairs)

of a certain brand of women’s shoes are given by the demand equation

p(x)= 100e^-0.0001x for 0 < x < 20,000

a.Find the revenue function, R. (Hint: R(x)= x(p(x)), since the

revenue function is the unit selling price at a demand level of x units

times the number of units demanded.)

b. Find the marginal revenue function, R'.

c. What is the marginal revenue when, x=10?

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