Ch. 4 - Applications of the Derivative

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 4 - Applications of the Derivative

Problem 4.1.87b

Chapter 4, Problem 4.1.87b

{Use of Tech} Every second counts You must get from a point P on the straight shore of a lake to a stranded swimmer who is 50 from a point Q on the shore that is 50 m from you (see figure). Assuming that you can swim at a speed of 2 m/s and run at a speed of 4 m/s, the goal of this exercise is to determine the point along the shore, x meters from Q, where you should stop running and start swimming to reach the swimmer in the minimum time. <IMAGE>

b. Find the critical point of T on (0, 50).

Verified step by step guidance

First, understand the problem setup: You need to minimize the time taken to reach the swimmer by running along the shore and then swimming. The swimmer is 50 meters from point Q, and you are initially 50 meters from Q along the shore.

Define the variables: Let x be the distance from point Q where you stop running and start swimming. The total distance you run is (50 - x) meters, and the distance you swim is 50 meters.

Express the time taken for each segment: The time to run is given by $ \frac{50 - x}{4} $ seconds, and the time to swim is $ \frac{50}{2} $ seconds. The total time T is the sum of these two times.

Formulate the function for total time T: $ T(x) = \frac{50 - x}{4} + \frac{50}{2} $. Simplify this expression to find T as a function of x.

Find the critical points: To find the critical points of T on the interval (0, 50), take the derivative of T with respect to x, set it equal to zero, and solve for x. This will give you the point where the time is minimized.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Optimization

Optimization in calculus involves finding the maximum or minimum values of a function. In this context, we need to minimize the total time taken to reach the swimmer by determining the optimal point along the shore to switch from running to swimming. This requires setting up a function that represents the total time as a function of the distance run and then finding its critical points.

Critical Points

Critical points are values in the domain of a function where the derivative is either zero or undefined. These points are essential in optimization problems as they help identify potential maxima or minima. In this scenario, finding the critical point of the time function T will allow us to determine the optimal distance to run before swimming.

Distance and Speed Relationships

Understanding the relationship between distance, speed, and time is crucial for solving this problem. The time taken to reach the swimmer consists of the time spent running and the time spent swimming, which can be expressed using the formula time = distance/speed. By breaking down the total distance into components based on the chosen point along the shore, we can formulate the time function needed for optimization.

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Textbook Question

Pen problems

b. A rancher plans to make four identical and adjacent rectangular pens against a barn, each with an area of 100 m² (see figure). What are the dimensions of each pen that minimize the amount of fence that must be used? <IMAGE>

Textbook Question

Values of related functions Suppose f is differentiable on (-∞,∞) and assume it has a local extreme value at the point x = 2, where f(2) = 0. Let g(x) = xf(x) + 1 and let h(x) = xf(x) + x +1, for all values of x.

b. Does either g or h have a local extreme value at x = 2? Explain.

Textbook Question

Cylinder in a cone A right circular cylinder is placed inside a cone of radius R and height H so that the base of the cylinder lies on the base of the cone.

b. Find the dimensions of the cylinder with maximum lateral surface area (area of the curved surface).

Textbook Question

{Use of Tech} Fixed points of quadratics and quartics Let f(x) = ax(1 -x), where a is a real number and 0 ≤ a ≤ 1. Recall that the fixed point of a function is a value of x such that f(x) = x (Exercises 48–51).

b. Consider the polynomial g(x) = f(f(x)). Write g in terms of a and powers of x. What is its degree?

Textbook Question

{Use of Tech} Demand functions and elasticity Economists use demand functions to describe how much of a commodity can be sold at varying prices. For example, the demand function D(p) = 500 - 10p says that at a price of p = 10, a quantity of D(10) = 400 units of the commodity can be sold. The elasticity E = dD/dp p/D of the demand gives the approximate percent change in the demand for every 1% change in the price. (See Section 3.6 or the Guided Project Elasticity in Economics for more on demand functions and elasticity.)

b. If the price is \$12 and increases by 4.5%, what is the approximate percent change in the demand?

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Textbook Question

{Use of Tech} Let f(x) = ln((x+1)/(x-1)) and g(x) = ln ((x+1)/(x-1)).

b. Sketch graphs of f and g to show that these functions do not differ by a constant.