Question

In Exercises 1-10, solve the problem analytically. Support your answer graphically. Finding Numbers The sum of two nonnegative numbers is 20\. Find the numbers if (a) the sum of their squares is as large as possible; as small as possible. (b) one number plus the square root of the other is as large as possible; as small as possible.

Short answer

For (a), the numbers are (0, 20) and (20, 0) for maximum sum of squares, (10, 10) for minimum sum of squares. For (b), the numbers for 'as large as possible' case are anything besides (0, 20) and (20, 0), and for 'as small as possible' are (0, 20) or (20, 0) for the sum of one number and the square root of the other.

Step-by-step solution

Case (a): Sum of Squares

Let's say one number is \( x \) then the other is \( 20 - x \). The sum of their squares is \( S = x^2 + (20 - x)^2 \). We need to find the extrema of \( S \). Differentiate \( S \) with respect to \( x \), we get \( S' = 2x - 40 + 2x = 4x - 40 \). Find when \( S' = 0 \), we get \( x = 10 \). When \( x = 0, 20 \), \( S = 400 \), when \( x = 10, S = 200 \). Thus the numbers are (0, 20) and (20, 0) for maximum, (10, 10) for minimum.

Case (b): One Number Plus Square Root of the Other

Let's continue with one number as \( x \) and the other is \( 20 - x \). Now the expression is \( F = x + \sqrt{20 - x} \). Differentiate \( F \) with respect to \( x \), we get \( F' = 1 - \frac{1}{2\sqrt{20-x}} \). Find when \( F' = 0 \), we get \( x = 10 \). When \( x = 0, 20, F = \sqrt{20} \). Thus the numbers for 'as large as possible' case are anything besides (0, 20) and (20, 0), and for 'as small as possible' are (0, 20) or (20, 0).

Graphical Representation

Graph the functions \( S = x^2 + (20 - x)^2 \) and \( F = x + \sqrt{20 - x} \) and indicate the maximum and minimum points. The minima and maxima found analytically should match the graphical representation. However, this being a mathematical exercise, it's not possible to depict them here.

Key concepts

Extrema of a Function

Understanding the extrema of a function plays a central role in calculus, particularly in the realm of optimization. Extrema refer to the maximum and minimum values that a function can take. In practical terms, this could mean finding the largest or smallest outcome of a given scenario, subject to certain constraints.

For instance, in the textbook exercise, we aim to maximize or minimize certain expressions given a condition that the sum of two nonnegative numbers is 20. To locate the extrema, we differentiate the expressions and set the derivative equal to zero. This method gives us critical points, which may correspond to local maxima or minima. By evaluating the function at these critical points and at the boundaries, we can determine which are indeed the extrema.

In more complex scenarios or functions with multiple variables, additional methods such as the Second Derivative Test or the use of Lagrange multipliers might be employed to identify these extreme values.

Derivative Applications

The application of derivatives is not limited to finding the slope of a tangent line but is also essential in solving real-world optimization problems. Derivatives tell us how a function is changing at any point and are used to find the rate of change of quantities. They enable us to understand the behavior of functions and to predict future values.

In our exercise, we use derivatives to determine how the sum of squares and the other function change as one of the numbers varies. By setting the derivative equal to zero, we identify the points where there is no change in slope—indicating a possible extremum. Derivative applications also extend to velocity, acceleration, economics for cost and revenue analysis, and physics for motion.

For students who may find this conception challenging, associative visual learning, such as graphing the functions or utilizing geometric interpretations, can greatly aid in comprehension.

Graphical Analysis in Calculus

Graphical analysis is a powerful tool in calculus for visualizing the behavior of functions and their derivatives. It complements analytical methods and provides a visual representation of the concepts discussed. By graphing functions, students can see the extrema, inflection points, and understand the concavity of functions, which elucidates many of the theoretical aspects of calculus.

In the context of our textbook problem, graphing provides a visual check for the work done analytically. Plotting the functions given in the problem would show the points corresponding to the maximum and minimum values identified earlier. This reinforces the understanding of the optimization process and aids in grasping concepts like continuity and differentiability. As an educator, it's beneficial to encourage students to use graphing tools or software to strengthen their analysis and reinforce their predictions made through analytical methods.

Solving Optimization Problems

Optimization problems are all about finding the best solution under a given set of conditions. They often involve maximizing or minimizing a quantity. To solve such problems in calculus, we typically follow a series of steps: first, we understand the problem and establish the function that needs to be optimized. Next, we identify constraints and express other variables in terms of a single independent variable, if possible.

After formulating the function, we find its derivative to locate the critical points. Checking these points, along with any endpoints, will lead us to the solution, revealing the optimized values. In our textbook example, the problem involves maximizing or minimizing the functions with respect to the sum of two numbers fixed at 20. The careful analytical process supplemented by graphical validation results in a complete understanding of where the optimized points lie.

For those still struggling with optimization, breaking it into these methodical steps and practicing with varied examples can be instrumental in improving their grasp. Additionally, the utilization of these steps across different functions and constraints is an excellent way to build proficiency in solving optimization problems.