Math 1325 Lab 3 Name ______

(Dr. Khoury) Show all workDue as scheduled

  1. Use implicit differentiation to find the slope(s) of the tangent line(s) to the graph of the circle at , and then find the equation(s) of the tangent line(s) at to the graph of the circle at .
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  1. A boat is being pulled towards a dock. If the rope is being pulled in at 3 feet per second, how fast is the distance between the dock and the boat decreasing when it is 30 feet from the dock?
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  1. Refer to #2, suppose the distance between the boat and the dock is decreasing at 3.05 feet per second. How fast is the rope being pulled in when the boat is 10 feet from the dock?

  1. Suppose two motorboats leave from the same point at the same time. If one travels north at 12 mph and the other travel east at 15 mph, how fast will the distance between them changing after 5 hours?
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  1. Suppose that for a company manufacturing car alarms, the cost and revenue equations are given by:

Where the production output in one month is x alarms. If the production is increasing at the rate of 100 alarms per month when production is 1000 alarms, find the rate of increase in:

  1. Cost
  1. Revenue
  1. Profit
  1. Suppose that for a company manufacturing calculators, the cost and revenue equations are given by:

Where the production output in one week is x calculators. If the production is increasing at the rate of 500 calculators per week when production is 6,000 calculators, find the rate of increase in:

  1. Cost
  1. Revenue
  1. Profit

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Spring 2017

  1. The price p (in dollars) and demand x for a product are related by .
  2. If the price is increasing at a rate of $2 per month when the price is $30, find the rate of change of the demand.
  1. If the demand is decreasing at a rate of 6 units per month when the demand is 150 units, find the rate of change of the price.
  1. The depreciation D, in dollars, of a company’s mainframe computer after t years is estimated by . What is the rate of depreciation in dollars per year after 10 years?

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Spring 2017

  1. Use L’Hospital’s rule to evaluate
  1. Use L’Hospital’s rule to evaluate
  2. Use L’Hospital’s rule to evaluate
  1. Use L’Hospital’s rule to evaluate

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Spring 2017

  1. An open box of maximum volume is to be made from a square piece of material, 24 inches on each side, by cutting squares from the corners and turning up the sides. Use calculus to find the dimensions of such box and its maximum volume.
  1. Find the location of any absolute and local extrema for the functions with graphs as follows.
  1. Find the location of any absolute and local extrema for the functions defined as follows over the specified domain. Use your graphing calculator to verify your answer.

a. ; b. ;

c. ; d. ;

c. ; d. ;

  1. A company has found that its weekly profit from the sale of x units of an auto part is given by . Production bottlenecks limit the number of units that can be made per week to no more than 150, while a long-term contract requires that at least 50 units be made each week. Find the maximum possible weekly profit that the firm can make.
  1. Find nonnegative numbers x and y that satisfy the given requirements.

a. and is as large as possible.

b. and is as small as possible.

c. and is as large as possible.

d. and is as large as possible.

  1. If the price charged for a candy bar is cents, then x thousand candy bars will be sold in a certain city.
  2. Find the total revenue in terms of x.
  1. Find the value of x that maximizes the total revenue.
  1. Find the maximum revenue.
  1. The sale of cassette tapes is very sensitive to price. If the price charged for a tape is dollars, then x thousand tapes will be sold in a certain city.
  2. Find the total revenue in terms of x.
  1. Find the value of x that maximizes the total revenue.
  1. Find the maximum revenue.
  1. A farmer has 1200m of fencing. He wants to enclose a rectangular field bordering a river, with no fencing needed along the river. Let x represent the width of the field. Find the value of x leading to the maximum area and the maximum area.
  1. A closed box with a square base is to have a volume of 16,000 . The material for the top and bottom of the box costs $3 per square centimeter, while the material for the sides costs $1.50 per square centimeter. Find the dimensions of the box that will lead to minimum total cost and the minimum total cost.
  1. Find by implicit differentiation for each of the following.

a. b.

c. d.

  1. Find the equation of the tangent line at the given point on each curve.

a. ; b. ;

  1. Find the equation of the tangent line at the given value of x on each curve.

a. ; b. ;

  1. The demand equation for a certain product is , where p is the price per unit in dollars and q is the number of units demanded.

a. Find b. Find

  1. If the revenue function and the cost function , where x is the daily production, find the following when 40 units are produced daily and the rate of change of production 10 units per day.
  2. The rate of change of revenue with respect to time.
  1. The rate of change of cost with respect to time

b. The rate of change of profit with respect to time

  1. A rock is thrown into a still pond. The circular ripples move outward from the point of impact of the rock so that the radius of the circle formed by a ripple increases at the rate of 2 feet pet minute. Find the rate at which the area is changing at the instant the radius is 4 feet.

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Spring 2017