Evaluate the derivative of the following functions.

f(t) = ln (sin^-1 t²)

Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.

f(x)= 1/(2x + 1); P (0,1)

Find and simplify the derivative of the following functions.

y = (3t−1)(2t−2)^-1

Consider the following cost functions.

c. Interpret the values obtained in part (b).

C(x) = 1000+0.1x, 0≤x≤5000, a=2000

Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.

f(x) = √(x - 1); P (2,1)

Consider the following cost functions.

b. Determine the average cost and the marginal cost when x=a.

C(x) = 500+0.02x, 0≤x≤2000, a=1000

Consider the following cost functions.

c. Interpret the values obtained in part (b).

C(x) = 500+0.02x, 0≤x≤2000, a=1000

Find and simplify the derivative of the following functions.

h(x) = (x − 1)(x³+ x² + x+1)

Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.

f(x) = √(x + 3); P (1,2)

Equations of tangent lines by definition (2)

b. Determine an equation of the tangent line at P.

f(x) = √x+3; P (1,2)

Calculate the derivative of the following functions.

y = sec(3x+1)

Demand and elasticity Based on sales data o​​ver the past year, the owner of a DVD store devises the demand function $D\left(p\right)=40-2p$ , where D(p) is the number of DVDs that can be sold in one day at a price of p dollars.

a. According to the model, how many DVDs can be sold in a day at a price of $10?

Based on sales data o​​ver the past year, the owner of a DVD store devises the demand function D(p) = 40 - 2p, where D(p) is the number of DVDs that can be sold in one day at a price of p dollars.

Find the elasticity function for this demand function.

Based on sales data o​​ver the past year, the owner of a DVD store devises the demand function D(p) = 40 - 2p, where D(p) is the number of DVDs that can be sold in one day at a price of p dollars.

For what prices is the demand elastic? Inelastic?

Calculate the derivative of the following functions.

y = csc e^x

Derivatives and tangent lines

a. For the following functions and values of a, find f′(a).

f(x) = x²; a=3

Derivatives and tangent lines

b. Determine an equation of the line tangent to the graph of f at the point (a,f(a)) for the given value of a.

f(x) = x²; a=3

Calculate the derivative of the following functions.

y = tan e^x

Find the derivative function f' for the following functions f.

f(x) =3x²+2x−10; a=1

Determine whether the following statements are true and give an explanation or counterexample.

c. It is impossible for the instantaneous velocity at all times a≤t≤b to equal the average velocity over the interval a≤t≤b.

A feather dropped on the moon On the moon, a feather will fall to the ground at the same rate as a heavy stone. Suppose a feather is dropped from a height of 40 m above the surface of the moon. Its height (in meters) above the ground after t seconds is s = 40−0.8t². Determine the velocity and acceleration of the feather the moment it strikes the surface of the moon.

Calculate the derivative of the following functions.

y = sin (4x³ + 3x +1)

Find the derivative function f' for the following functions f.

f(x) = √3x+1; a=8

Derivatives and tangent lines

b. Determine an equation of the line tangent to the graph of f at the point (a,f(a)) for the given value of a.

f(x) = 1/ √x; a= 1/4

Calculate the derivative of the following functions.

y = csc (t² + t)

Derivatives and tangent lines

a. For the following functions and values of a, find f′(a).

f(x) = 1/ x²; a= 1

Find an equation of the line tangent to the graph of f at (a, f(a)) for the given value of a.

f(x) = √x+2; a=7

Find and simplify the derivative of the following functions.

f(x) = 3x^-9

Find the derivative function f' for the following functions f.

f(x) = 2/3x+1; a= -1

Find an equation of the line tangent to the graph of f at (a, f(a)) for the given value of a.

f(x) = 1/x; a= -5