**Section I:**

**The Derivative as a Rate of Change**

In case of a linear function , the graph is a straight line and the slope m measures the steepness of the line by giving the rate of climb of the line, the rate of change of y with respect to x.

As x changes from to , y changes m times as much:

Thus, the slope gives the change in y per unit change in x.

In more general case of differentiable function , the difference quotient

, where

give the **average rate of change of y (or f)** with respect to x. The limit as h approaches zero is the derivative , which can be interpreted as the *instantaneous rate of change of f *with respect to x. Since, the graph is a curve, the rate of change of y can vary from point to point.

*Velocity and Acceleration:*

Suppose that an object is moving along a straight line and that, for each time t during a certain time interval, the object has location/position . Then, at time the position of the object is and is the change in position that the object experienced during the time period t to . The ratio

gives the average velocity of the object during this time period. If

exists, then gives the instantaneous rate of change of position with respect to time. This rate of change of position is called the *velocity* of the object. If the velocity function is itself differentiable, then its rate of change with respect to time is called the *acceleration*; in symbols,

The speed is by definition the absolute value of the velocity: speed at time t is

If the velocity and acceleration have the same sign, then the object is speeding up, but if the velocity and acceleration have opposite signs, then the object is slowing down.

A sudden change in acceleration is called a *jerk. Jerk *is the derivative of acceleration. If a body’s position at the time t is , the body’s jerk at time t is

*Differentials*

Let be a differentiable function. Let . The difference is called the increment of f from x to , and is denoted by .

The product is called the differential of f at x with increment h, and is denoted by

The change in f from x to can be approximated by :

**Tangent and Normal**

Let be the equation of a curve, and let be a point on it. Let PT be the tangent, PN the normal and PM the perpendicular to the x-axis.

The slope of the tangent to the curve at P is given by

Thus, the equation of the tangent to the curve at is

Since PM is perpendicular to PT, it follows that if , the slope of PN is

Hence, the equation of the normal to the curve at is

The equation of the normal parallel to the x-axis is , that is, when . The length of the tangent at is PT, and it is equal to

The length of the normal is PN and it is equal to

If the curve is represented by and , that is, parametric equations in t, then

where and . In this case, the equations of the tangent and the normal are given by

and respectively.

**The Angle between Two Curves**

The angle of intersection of two curves is defined as the angle between the two tangents at the point of intersection. Let and be two curves, and let be their point of intersection. Also, let and be the angles of inclination of the two tangents with the x-axis, and let be the angle between the two tangents. Then,

*Example 1:*

Write down the equations of the tangent and the normal to the curve at the point .

*Solution 1:*

.

Hence, the equation of the tangent at is given by and the equation of the normal is .

**Rolle’s Theorem and Lagrange’s Theorem:**

*Rolle’s Theorem:*

Let be a function defined on a closed interval such that (i) f(x) is continuous on , (ii) f(x) is derivable on , and (iii) f(a) = f(b). Then, there exists a such that .

*For details, the very beautiful, lucid, accessible explanation in Wikipedia:*

https://en.wikipedia.org/wiki/Rolle%27s_theorem

*Lagrange’s theorem:*

Let be a function defined on a closed interval such that (i) is continuous on , and (ii) is derivable on . Then, there exists a such that

*Example 2:*

The function satisfies the conditions of Rolle’s theorem on the interval , as the logarithmic function and are continuous and differentiable functions and .

The conclusion of Rolle’s theorem is given at , for which .

*Rolle’s theorem for polynomials:*

If is any polynomial, then between any pair of roots of lies a root of .

**Monotonicity:**

A function defined on a set D is said to be non-decreasing, increasing, non-increasing and decreasing respectively, if for any and , we have , , and respectively. The function is said to be monotonic if it possesses any of these properties.

For example, is an increasing function, and is a decreasing function.

*Testing monotonicity:*

Let be continuous on and differentiable on . Then,

(i) for to be non-decreasing (non-increasing) on it is necessary and sufficient that () for all .

(ii) for to be increasing (decreasing) on it is sufficient that () for all .

(iii) If for all x in , then f is constant on .

*Example 3:*

Determine the intervals of increase and decrease for the function .

*Solution 3:*

We have , and for any value of x, . Hence, f is increasing on . QED.

*The following is a simple criterion for determining the sign of *:

If , then iff or ;

if and only if

**Maxima and Minima:**

A function has a local *maximum* at the point if the value of the function at that point is greater than its values at all points other than of a certain interval containing the point . In other words, a function has a maximum at if it is possible to find an interval containing , that is, with , such that for all points different from in , we have .

A function has a local minimum at if there exists an interval containing such that for and .

One should not confuse the local maximum and local minimum of a function with its largest and smallest values over a given interval. The local maximum of a function is the largest value only in comparison to the values it has at all points sufficiently close to the point of local maximum. Similarly, the local minimum is the smallest value only in comparison to the values of the function at all points sufficiently close to the local minimum point.

The general term for the maximum and minimum of a function is *extremum*, or the extreme values of the function. A necessary condition for the existence of an extremum at the point of the function is that , or does not exist. The points at which or does not exist, are called *critical points.*

**First Derivative Test:**

(i) If changes sign from positive to negative at , that is, for and for , then the function attains a local maximum at .

(ii) If changes sign from negative to positive at , that is, for , and for , then the function attains a local minimum at .

(iii) If the derivative does not change sign in moving through the point , there is no extremum at that point.

**Second Derivative Test:**

Let f be twice differentiable, and let c be a root of the equation . Then,

(i) c is a local maximum point if .

(ii) c is a local minimum point if .

However, if , then the following result is applicable. Let (where f^{r} denotes the rth derivative), but .

(i) If n is even and , there is a local maximum at c, while if , there is a local minimum at c.

(ii) If n is odd, there is no extremum at the point c.

**Greatest/Least Value (Absolute Maximum/Absolute Minimum):**

Let f be a function with domain D. Then, f has a greatest value (or absolute maximum) at a point if for all x in D and a least value (or absolute minimum) at c, if for all x in D.

If f is continuous at every point of D, and , a closed interval, the f assumes both a greatest value M and a least value m, that is, there are such that and , and for every .

*Example 4:*

a) , with domain . This has no greatest value; least value at

b) with domain . This has greatest value at and least value at .

c) with domain . This has greatest value at and no least value.

d) with domain . This has no greatest value and no least value.

Some other remarks:

The greatest (least) value of continuous function on the interval is attained either at the critical points or at the end points of the interval. To find the greatest (least) value of the function, we have to compute its values at all the critical points on the interval , and the values of the function at the end-points of the interval, and choose the greatest (least) out of the values so obtained.

*We will continue with problems on applications of derivatives later,*

Nalin Pithwa.