Calculus Answers

Suppose a car is traveling with variable velocity for one hour, and that
• At some point between 0 and 12 minutes, the car is moving with velocity 40 mph.
• At some point between 12 and 30 minutes, the car is moving with velocity −10 mph.
• At some point between 30 and 40 minutes, the car is moving with velocity 20 mph.
• At some point between 40 and 60 minutes, the car is moving with velocity 0 mph.
Use Riemann sums to approximate the total displacement of the car from its initial position.

Estimate the area under the graph of ƒ(x)=√(x+1) from x=−1 to x=3 using four rectangles and

- right endpoints,
- left endpoints,
- midpoints.

Use sigma notation to represent the area under the curve y=1/x​ on the interval [1, 4] using left hand approximation and 6 subintervals.

a. A manufacturer’s marginal revenue function is MR=105-x-0.3x^2. Find the increase in
the manufacturer’s total revenue if production is increased from 10 to 20 units.

b. A firm has a marginal revenue given by MR=(3/2x+7)-(1/20), where x is the output. Find
the corresponding demand function.

c. Given that the elasticity of demand for a commodity is given by e_xp=3-2p, where p denotes the price per unit of the commodity. find the demand function x.

The sides of a square are increasing at a rate of 10 cm/sec. How fast is the area enclosed by the square increasing when the area is 150 cm2.

a. A manufacturer’s marginal revenue function is MR=105-x-0.3x^2. Find the increase in
the manufacturer’s total revenue if production is increased from 10 to 20 units.

b. A firm has a marginal revenue given by MR=(3/2x+7)-(1/20), where x is the output. Find
the corresponding demand function.

c. Given that the elasticity of demand for a commodity is given by e_xp=3-2p, where p denotes the price per unit of the commodity. find the demand function x.

a. A manufacturer’s marginal revenue function is
2 MR x x    105 0.3
. Find the increase in
the manufacturer’s total revenue if production is increased from 10 to 20 units. 4 marks
b. A firm has a marginal revenue given by
3 1
2 7 20
MR
x
 

, where
x
is the output. Find
the corresponding demand function. 6 marks
c. Given that the elasticity of demand for a commodity is given by
3 2 ,
xp
e p  
where
p
denotes the price per unit of the commodity. find the demand function
x

Prove that

(v.del)v = 1/2 del v^2 - v×(del×v)

v is a vector

v dv/dx = -9x
write v as a function of x