# The n and C are constants. But what

The use of arithmetic in farming is essential and frequently expects capacities to portray and foresee different phenomenon’s. One of them incorporates consistently changing under fluctuating rates, and this requires calculus. Calculus fundamentally is a division of mathematics which manages the instantaneous rates in change. There are two wide applications of calculus: differential calculus (which is just known as separation or differentiation) and the other one is essential analytics (otherwise Calculus, unfortunately, has a reputation of being difficult and abstract to most people (agriculturists included). Most of us learn by rote the various expressions for differentiation and integration. The leading into what led to the development of using more analytic standpoint in calculus in agriculture was starting to develop around the time of the green revolution. called anti-differentiation or integration). RB2

Constant rate of change : Application of mathematics in agriculture often requires mathematical functions to describe and predict a phenomenon that is continuously changing under varying rates, and this requires the use of calculus. Calculus is a branch of mathematics that deals with instantaneous rates of change. And there are two broad applications of calculus: differential calculus (or simply known as differentiation) and the other integral calculus (also known as integration or anti-differentiation).

Calculus, unfortunately, has a reputation of being difficult and abstract to most people (agriculturists included). Most of us learn by rote the various expressions for differentiation and integration, including

and its counterpart

where both n and C are constants. But what is the point of calculus? More specifically, how do we use or apply calculus in agriculture?

Consider a hypothetical growing crop bean. Table 2.1 shows that the plant’s growth rate remained constant at 10 g day-1 from day 10 to 20. This means that in this period the plant would consistently gain weight by 10 g per day. So after ten days, the plant must gain (10 g day-1 ´ 10 days) or 100 g. Hence, provided the plant is growing at a constant rate, there is a simple relationship between the cumulative (total) weight change (DW) and the rate of growth (wr):

where Dt is the time interval. In Table 2.1, we can confirm that in ten days from day 10 to 20 (Dt = 10) the plant weight increased by 100 g.

And if we plot the growth rate (which is constant) against time, we will get a horizontal line at 10 g day-1 (Fig. 2.1a). Furthermore, the area under this line is actually the cumulative weight change. As mentioned earlier, the cumulative weight change is the growth rate multiplied by the time interval. The area under the line in Fig. 2.1a is a rectangle with a height of 10 g day-1 (i.e., the growth rate) and a width of (20 – 10) days (i.e., the time interval). Consequently, this gives the area under the line from day 10 to 20 as the area of the rectangle:

This cumulative change in weight of 100 g from day 10 to 20 agrees with our previous calculation and the data in Table 2.1. In short, the cumulative change in weight is the area under the line (or curve) for the graph of rate of weight change against time.

Table 2.1. Measured plant growth (constant growth rate)

Day

Growth rate (g day-1)

Plant weight (g)

10

10

300

11

10

310

12

10

320

13

10

330

14

10

340

15

10

350

16

10

360

17

10

370

18

10

380

19

10

390

20

10

400

(a)

(b)

Fig. 2.1. Plotted constant growth rate and plant weight against time

Since the growth rate remained constant, the plant weight must increase linearly with time (Fig. 2.1b). And if we take the slope of this linear line, we will obtain 10 g day-1, which is the rate of weight change or growth. Recall that for a linear line, its slope (m) is calculated as

where (x1, y1) and (x2, y2) are two pairs of points along the linear line. From Table 2.1, let us take the two pairs of points as (15, 350) and (20, 400), so that we determine the slope of the line as

In other words, the instantaneous rate of weight change is the slope of the line (or curve) for the graph of plant weight against time.

To generalise: the slope of a curve gives us the instantaneous rate of change, whereas the area under a rate of change curve gives us the cumulative change. As we will later discuss, the slope of a curve and the area under a curve are concerns of differential calculus and integral calculus, respectively.

1.1 Variable rate of change

In the previous section, calculations were simple because the rate of plant growth was constant. But when the rates of change are not constant, the use of calculus then becomes essential. Consider a second example using the same hypothetical crop but it grew instead at a variable (non-constant) rate.

Table 2.2 shows the measured daily plant weight from day 10 to 20. Using the data in Table 2.2, we can show that the plant weight is related to time by the following quadratic function:

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where f (t) is the weight of the plant (g) at time t (day). For example:

at day 10:

at day 11:

…

at day 20:

all of which agrees with the tabulated data in Table 2.2.

Plotting the plant weight against time shows that the plant weight increased with time in a non-linear manner (Fig. 2.2). This shows that the growth rate of the plant is not constant, in contrast to the first example (Fig. 2.1b). As mentioned in the previous section, finding the slope of a curve will give us the instantaneous rate of change. When the growth rate is constant, the plant weight will increase linearly with time. This increase must be linear because a linear line has a constant slope at all its points (hence, a constant rate of change).

Table 2.2. Measured plant weight which increases at a variable rate

Day

Plant weight (g)

10

300

11

322

12

346

13

372

14

400

15

430

16

462

17

496

18

532

19

570

20

610

Fig. 2.2. Plotted plant weight against time (variable growth rate)

But when the growth rate is variable, as in this second example, the weight of the plant will increase in a non-linear manner with time. Because this curve is non-linear, its slope will vary from point-to-point along the curve, in contrast to a linear curve. Finding the slope of a curve at, say, point (a, b) is, by definition, finding the slope of the tangent line at (a, b). But what exactly is a tangent line?

Fig. 2.3 shows point P on two curves. We have drawn an enlarged version of the box around each point P. Notice that the portion of each curve within the boxed region looks almost straight. With increasing magnification, the portion of the curve near P becomes increasingly more exactly like a straight line. This straight line is called the tangent line to the curve at point P.

Fig. 2.3. Tangent line to the curve at point P

The fundamental idea to determine the slope of the tangent line at a point P is to approximate the tangent line very closely by secant lines. A secant line at P is a straight line passing through P and a nearby point Q on the curve (Fig. 2.4). Now, suppose that point P is (x, f (x)), and that point Q is h horizontal units away from point P so that point Q is located at (x + h, f (x + h)). Consequently, the slope of the secant line through points P and Q is

To let the slope of secant line approach the slope of tangent line, we move point Q increasingly closer to point P, thereby h becomes increasingly smaller (but h is never zero). In other words, by taking h sufficiently small, the slope of the secant line can be taken as the slope of the tangent line with the desired accuracy. Mathematically, we write this as

where h®0 means h approaches but never reaches zero, and f’ (a) is the derivative of the function f (x) at x = a. In other words, f’ (a) is the slope of the tangent line at x = a. As mentioned in the previous chapter, the derivative of the function f (x) is sometimes written as dy/dx, df (x)/dx or D(x).

Fig. 2.4. Tangent line to the curve at point P approximated by the secant line through points P and Q

Let us return to the example where the plant growth rate is variable. What is the rate of plant growth at, say, day 15? To do this, we need to find the slope of the tangent line at t = 15. In other words, we differentiate the plant weight function f (t) = t2 + t + 190 at t = 15:

where we see that as h®0, f’ (15) approaches 31. Consequently, we can take 31 g day-1 as the plant growth rate at day 15. Likewise, to determine the growth rate at day 20 is

where the growth rate at day 20 is 41 g day-1. To generalise, the growth rate of the plant at any given time t is:

So the derivative of the plant weight function f (t) = t2 + t + 190 gives the growth rate function as f’ (t) = 2t + 1. Table 2.3 expands on the earlier Table 2.2 by showing the measured daily plant weight as well as the calculated daily growth rates (i.e., 2t + 1) from day 10 to 20.

Table 2.3. Measured plant weight and its variable (calculated) rate of increase

Day

Growth rate (g day-1)

Plant weight (g)

10

21

300

11

23

322

12

25

346

13

27

372

14

29

400

15

31

430

16

33

462

17

35

496

18

37

532

19

39

570

20

41

610

Fig. 2.5 shows that the growth rate increased linearly with time, where the line is described by the function 2t + 1. We saw in the previous section that if we took the area under the rate of change curve, we would obtain the cumulative change. In Fig. 2.5, the area under the curve from day 10 to 20 is the combined area of the triangle (Area A) and rectangle (Area B):

which agrees with our measured weight gain of 310 g from day 10 to 20 (Table 2.3).

Fig. 2.5. Plotted variable growth rate against time

Finding the area under the curve is the same as integration, and integration is the converse of differentiation. When we take the slope of the function f, we interpret it as its rate of change. But if we wish to determine the cumulative change (that is, how much change had occurred), we take the anti-derivative of the rate of change function f’; that is, we integrate the rate function f’. In short, by integrating the function that describes the rate of change curve, it gives us the cumulative change; that is,

In our first example, the instantaneous rate of change was constant at 10 g day-1. Hence, integrating

gives us the cumulative change of 100 g from day 10 to 20 (Table 2.1). But in our second example, the rate of growth varied according to the function 2t + 1 so integrating this function

gives us a cumulative change of 310 g in the same ten-day period (Table 2.3).

1.2 Integration as a summation and the mean value of a function

In Fig. 2.6, the area under the curve ABCD can be approximated by dividing this area into several rectangular strips, where each strip has a width of dx (the symbol ‘d’ is interpreted as “very small”). The area of each strip is approximately equal to f (x)×dx (i.e., height ´ width for a rectangle area). And the sum area for all the strips gives the approximate area of ABCD, or

2.1

Notice that the last term f (b)×dx is the area of the last rectangle which lies just outside the region ABCD, and that, in Fig. 2.6, the areas of the rectangles tend to underestimate the area under the curve. The area of the first rectangle, for example, underestimates the area APRD by the area APQ.

Fig. 2.6. Approximating the area under the curve by a series of equal width rectangles

Eq. 2.1, however, becomes increasingly more accurate if we take the width of each strip to be increasingly smaller and so increasing the number of strips to cover ABCD. In other words, as dx approaches zero (dx ® 0), the accuracy of Eq. 2.1 increases. Mathematically, we write this as

We have learnt that the area under a curve ABCD is determined exactly by integrating the function f (x) from x = a to x = b:

so this means that

2.2

Eq. 2.2 is important as it tells us that an integral can be approximated by a summation, provided that the area under the curve can be divided into a large number of narrow strips to give the desired accuracy.

Let us further determine the mean (average) of the function f (x) from x = a to x = b (Fig. 2.6). Recall that the mean of n quantities (y1, y2, …, yn) is:

In the same way, the mean of the function from a to b is given by

Since (b – a) = (n – 1)×dx, and as n ® ¥, dx ® 0:

where earlier in Eq. 2.2 we showedRB4 that .

RB1Introduction needs to be explained that why you have chosen topic. Please next state the aim

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RB3Please show how you got the function.

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