Derivative and the Maximum Area Problem

Note: This is the third part of the Derivative Concept Series. The first part is The Algebraic and Geometric Meaning of Derivative and the second part is Derivative in Real Life Context.

Introduction

The computation of derivative is often seen in maximum and minimum problems.  In this article, we will discuss why do we get the derivative of a function and equate it to 0 when we want to get its maximum or minimum. To give you a concrete example, let us consider the problem below.

Find the maximum area a rectangle with perimeter 10 units.

Without using calculus, we can substitute values for the rectangle’s length, compute for its width and its corresponding area. If we set the interval to 0.5, then we can come up with the table shown in Figure 1.

Figure 1 - Table showing the length, width, and area of a rectangle with perimeter 10.

Looking at the table above, we can observe that a rectangle of length of 2.5, a square, has the maximum area. If we have prior calculus  knowledge, however, we know that whatever the value of our perimeter, a square having the given perimeter will always have the maximum area.

Using elementary algebra, if we let x be the width of our rectangle, it follows that the length is 5-x. Let f(x) be the area of the rectangle. In effect, the area of the rectangle is described by the equation f(x) = 5x - x^2. We want to maximize the area, which implies that we want to find the maximum value of f(x).

Figure 2 – A rectangle with Perimeter 10 and width x units.

In elementary calculus, to compute for the maximum value of f(x), we get its derivative, which is equal to 5 - 2x, which we will denote f'(x). We then equate the f'(x) to 0 resulting to the equation 5-2x=0 \Rightarrow x = 2.5 which is exactly the maximum value in the table above.

Derivative and Equation to 0

In the article the Algebraic and Geometric Meaning of Derivative, we have learned that the derivative of a function is the slope of the line tangent to that function at a particular point. From elementary algebra, we also have learned the properties of slopes. If a line is rising to the right, the slope is greater than 0; if the line is rising to the to the left, then the slope is less than 0. We have also learned that a horizontal line has slope 0 and the vertical line has an undefined slope.

Figure 3 – Properties of slope of a straight line.

In the problem above, we calculated by getting the derivative (the slope of the line tangent to a function at a particular point) and equate it to 0. But a line with slope 0 is a horizontal line. In effect, we are looking for a horizontal tangent of f(x) = 5x-x^2. To give a clearer picture let us look at the graph of f(x) = 5x - x^2.

Figure 4 – Tangent lines of 5x – x2.

From the graph it is clear that the maximum point of the function is where the tangent line (red line) horizontal. In fact, there are only three possible cases that tangent line could be horizontal as shown in Figure 5: first, the minimum of a function (blue graph); second, the inflection point (red graph); and the third is the maximum of the function (green graph).

It should also be noteworthy to say that all the ordered pairs (length, area) or(width, area) in Figure 1 will be on the blue curve in Figure 4.

Figure 5 – Cases of a graph where the tangent is horizontal.

The derivative has many applications and it is seen in many topics in calculus.  In the next Derivative Tutorial, we are going to discuss how the derivative is used in other context.

Summary

  • The derivative of a function is the slope of the line tangent to a function at a particular point.
  • The horizontal line has slope zero.
  • In solving maxima and minima problems, we get the derivative of a function and equate to zero to get the minimum or maximum. We do this because geometrically, we want to get the line tangent to a function at a particular point that is horizontal.

Derivative in Real Life Context

Note: This article is the second part of the derivative concept series. The first part is  The Algebraic and Geometric Meaning of Derivative and the third part is Derivative and the Maximum Area Problem.

If we change the labels of the Figure 2 in the The Algebraic and Geometric Meaning of Derivative article – its x-axis to time and the y-axis to distance – the graph of the secant line is the difference in distance y_2 - y_1 =  100 km – 50 km over the difference  in timex_2 - x_1 = 10:00 – 8:00.  This is shown in the figure below.  This is equivalent to 50 km/2 hrs or 25 km/hr. From the computation above, it is clear that the interpretation of the slope of the secant line is the total distance over the total time or the average speed.

On the other hand, the slope of the tangent line is the speed of the bicycle at exactly 4 o’clock. At exactly 4:00 o’clock the bicycle was traveling 50 km per hour, a lot faster than its average speed. Now, this is reasonable because in real life, the speed of travel is not always constant.  The slope of the tangent line or the speed of travel at a particular point is called the instantaneous speed.

We have also observed from the previous article that as Q approaches P, the secant line’s approximation of the tangent line becomes better and better. This means that as Q approaches P, the average speed becomes closer and closer to the instantaneous speed at P.

The Algebraic and Geometric Meaning of Derivative

Note: This is the first part of the Derivative Concept Series. The second part is Derivative in Real Life Context and the third part is Derivative and the Maximum Area Problem.

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If we want to get the slope of a line, we need two points. Suppose the points have coordinates (x_1,y_1) and (x_2,y_2), we have learned that the slope is described by the formula \displaystyle\frac{y_2-y_1}{x_2-x_1}.

In Figure 1, we have line l tangent to the function f at point P where the coordinates of P are (x,f(x)).  The problem that gave birth to calculus is getting the slope of this tangent line. There is, however, a problem.  We need two points to compute for the slope but we have only one point.

Note that the word tangent in this problem is different from the definition of tangent on a circle because it is clear that line l will intersect the graph in more than one point.

Figure 1 - Line l tangent to the function f at point P.

Using the concept of limits we can remedy this problem. First, we create point Q with x coordinate h units to the right of the x-coordinate of P. We then draw line PQ, a secant line to the function f.

Figure 2 - A secant line is drawn through P.

In effect, the coordinates of Q would be (x+h, f(x+h)) and it is clear that the slope of the secant line PQ is described by the formula

\displaystyle\frac{f(x+h)-f(x)}{x+h - x} = \frac{f(x+h)-f(x)}{h}

If we want to approximate the slope of the tangent line, it is reasonable that we move Q towards P with P fixed. Click here to explore the diagram above using GeoGebra.

From the GeoGebra exploration above, if we move Q towards P, we observe the following:

1.)    The value of h approaches 0.

2.)    The inclination of the secant line approaches the inclination of the tangent line.

3.)    The slope of the secant line approaches the slope of the tangent line.

4.)    If point Q coincides with point P, then the slope of the secant line and is equal to the slope of the tangent line.

If we let m be the slope of the secant line andf'(x) be the slope of the tangent line, focusing on observations 1 and 4, we can say the following equivalent statements:

  • The limit of the slope of the secant line m as Q approaches P is equal to f'(x).
  • The limit of the slope of the secant line m as h approaches 0 is equal to f'(x).
  • The limit of \displaystyle \frac{f(x+h)-f(x)}{h} as h approaches 0 is equal to f'(x).

Using the limit notation, we can say that

f'(x) = \lim_{h \to 0} \displaystyle\frac{f(x+h)-f(x)}{h}

From the above discussion, we can see that the derivative of a function at  a particular point is the slope of the line tangent to that function at that particular point.

In the next post, we will discuss the meaning of derivative in real life situations.

An Intuitive Introduction to Limits

Limits is one of the most fundamental concepts of calculus. The foundation of calculus was not entirely solid during the time of Leibniz and Newton, but later developments on the concept, particularly the \epsilon-\delta definition by Cauchy, Weierstrass and other mathematicians established its firm foundation. In the discussion below, I shall introduce the concept of limits intuitively as it appears in common problems. For a more rigorous discussion, you can read the post article titled “An extensive explanation about the \epsilon-\delta definition of limits”.

Circumference and Limits

If we are going to approximate the circumference of a circle using the perimeter of an inscribed polygon, even without computation, we can observe that as the number of sides of the polygon increases, the better the approximation. In fact, we can make the perimeter of the polygon as close as we please to the circumference of the circle by choosing a sufficiently large number of sides.  Notice that no matter how large the number of sides our polygon has, its perimeter will never exceed or equal the circumference of the circle.

Introduction to Limits

Figure 1 – As the number of side of the polygons increases, its perimeter gets closer to the circumference of the circle.

 

In a more technical term, we say that the limit of the perimeter of the inscribed polygon as the number of its sides increases without bound (or as the number of sides of the inscribed polygon approaches infinity) is equal to the circumference of the circle.  In symbol, if we let n be the number of sides of the inscribed polygon, P_n be the perimeter of a polygon with n sides, and C be the circumference of the circle, we can say that the limit of P_n as n \to \infty is equal to C. Compactly, we can write \lim_{n \to \infty} P_n = C.

Functions and Limits

Consider the function f(x) = \frac{1}{x} where x is a natural number. Calculating the values of the function using the first 20 natural numbers and plotting the points in the xy-plane, we arrive at the table and the graph in Figure 2.

Introduction to Limits

Figure 2 – As x increases, f(x) gets closer and closer to 0.

First, we see that as the value of x increases, the value of f(x) decreases and approaches 0. Furthermore, we can make the value of f(x) as close to 0 as we please by choosing a sufficiently large x. We also notice that no matter how large the value of x is, the value of f(x) will never reach 0.

Hence, we say that the limit of f(x) = \frac{1}{x} as the value of x increases without bound is equal to 0, or equivalently the limit of f(x) = \frac{1}{x} as x approaches infinity is equal to 0. In symbol, we write the limit of f(x) \to \infty as x \to 0 or more compactly the \lim_{x \to \infty} \frac{1}{x} = 0.

Tangent line and Limits

Recall that the slope of a line is its “rise” over its “run”. The formula of slope m of a line is m = \displaystyle\frac{y_2 - y_1}{x_2 - x_1}, given two points with coordinates (x_1,y_1) and (x_2,y_2).  One of the famous ancient problems in mathematics was the tangent problem, which is getting the slope of a line tangent to a function at a point.  In the Figure 3, line n is tangent to the function f at point P.

Figure 3 – Line n is tangent to the function f at point P.

If we are going to compute for the slope of the line tangent line, we have a big problem because we only have one point, and the slope formula requires two points.  To deal with this problem, we select a point Q on the graph of f, draw the secant line PQ and move Q along the graph of f towards P. Notice that as Q approaches P (shown as Q' and Q''), the secant line gets closer and closer to the tangent line. This is the same as saying that the slope the secant line is getting closer and closer to the slope of the tangent line. Similarly, we can say that as the distance between the x-coordinates of P and Q is getting closer and closer to 0, the slope of the secant line is getting closer and closer to the slope of the tangent line.

Figure 4 – As point Q approaches P, the slope of the secant line is getting closer and closer to the slope of the tangent line.

If we let h be the distance between the x-coordinates of P and Q, m_s be the slope of the secant line PQ and m_t be the slope of the tangent line, we can say that the limit of the slope of secant line as h approaches 0 is equal to the slope of the tangent line. Concisely, we can write \lim_{h \to 0}m_s = m_t.

Area and Limits

Another ancient problem is about finding the area under a curve as shown in the leftmost graph in Figure 5. During the ancient time, finding the area of a curved plane was impossible.

 

Introduction to Limits

Figure 5 – As the number of rectangles increases, the sum of the area of the rectangles is getting closer and closer to the area of the bounded plane under the curve.

We can approximate the area above in the first graph in Figure 5 by constructing rectangles under the curve such that one of the corners of the rectangle touches the graph as shown in the second and third graph in Figure 5. We can see that as we increase the number of rectangles, the better is our approximation of the area under the curve. We can also see that no matter how large the number of rectangles is, the sum its areas will never exceed (or equal) the area of the plane under the curve. Hence, we say that as the number of rectangles increases without bound, the sum of the areas of the rectangles is equal to the area under the curve; or the limit of the sum of the areas of the rectangles as the number of rectangles approaches infinity is equal to the area of the plane under the curve.

If we let A be the area under the curve, S_n be the sum of the areas of n rectangles, then we can say that the limit of S_n as n approaches infinity is equal to A. Concisely, we can write \lim_{n \to\infty} S_n = A.

Numbers and Limits

We end with a more familiar example usually found in books. What if we want to find the limit of 2x + 1 as x approaches 3?

To answer the question, we must find the value 2x + 1 where x is very close to 3. Those values would be numbers that are very close to 3 – some slightly greater than 3 and some slightly less than 3. Place the  values in a table we have

Figure 6 – As x approaches 3, 2x + 1 approaches 7.

From the table, we can clearly see that as the value of x approaches 3, the value of 2x + 1 approaches 7.  Concisely, we can write the \lim_{x \to 3} 2x + 1 =7.

Mr. Jayson Dyer, author of The Number Warrior has another excellent explanation on the concept of limits in his blog Five intuitive approaches to teaching the infinitely small.

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The area under the curve problem and the tangent problem are the ancient problems which gave birth to calculus. Calculus was independently invented by Gottfried Leibniz and Isaac Newton in the 17th century.

 

 

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