Time is Life.

Money lost can come back, but time lost can never come back. Time is more valuable than money. (Bertrand Russell).

Mathematics demystified

November 27, 2018 – 10:40 pm

Time is Life.

Money lost can come back, but time lost can never come back. Time is more valuable than money. (Bertrand Russell).

November 27, 2018 – 10:36 pm

The object of pure Physics is the unfolding of the laws of the intelligible world; the object of pure Mathematic that of unfolding the laws of human intelligence. — J. J. Sylvester.

*In my opinion, for example, Boole’s Laws of (Human) Thought. *

November 27, 2018 – 10:07 pm

Newton’s patience was limitless. Truth, he said much later, was the offspring of silence and meditation. And, he said, I keep the subject constantly before me and wait till the first dawnings open slowly, by little and little into a full and clear light.

November 13, 2018 – 6:26 am

**Question 1.**

If the point on , where , where the tangent is parallel to has an ordinate , then what is the value of ?

**Question 2:**

Prove that the segment of the tangent to the curve , which is contained between the coordinate axes is bisected at the point of tangency.

**Question 3:**

Find all the tangents to the curve for that are parallel to the line .

**Question 4:**

Prove that the curves , where , and , where is a differentiable function have common tangents at common points.

**Question 5:**

Find the condition that the lines may touch the curve .

**Question 6:**

Find the equation of a straight line which is tangent to one point and normal to the point on the curve , and .

**Question 7:**

Three normals are drawn from the point to the curve . Show that c must be greater than 1/2. One normal is always the x-axis. Find c for which the two other normals are perpendicular to each other.

**Question 8:**

If and are lengths of the perpendiculars from origin on the tangent and normal to the curve respectively, prove that .

**Question 9:**

Show that the curve , and is symmetrical about x-axis and has no real points for . If the tangent at the point t is inclined at an angle to OX, prove that . If the tangent at meets the curve again at Q, prove that the tangents at P and Q are at right angles.

**Question 10:
**

Find the condition that the curves and intersect orthogonality and hence show that the curves and also intersect orthogonally.

More later,

Nalin Pithwa.

October 23, 2018 – 9:43 pm

*Slightly difficult questions, I hope, but will certainly re-inforce core concepts:*

- Prove that the segment of the tangent to the curve which is contained between the co-ordinate axes, is bisected at the point of tangency.
- Find all tangents to the curve for that are parallel to the line .
- Prove that the curves , where and , where is a differentiable function, have common tangents at common points.
- Find the condition that the lines may touch the curve .
- If and are lengths of the perpendiculars from origin on the tangent and normal to the curve respectively, prove that .
- Show that the curve , is symmetrical about x-axis and has no real points for . If the tangent at the point t is inclined at an angle to OX, prove that . If the tangent at meets the curve again at Q, prove that the tangents at P and Q are at right angles.
- A tangent at a point other than on the curve meets the curve again at . The tangent at meets the curve at and so on. Show that the abscissae of form a GP. Also, find the ratio of area .
- Show that the square roots of two successive natural numbers greater than differ by less than .
- Show that the derivative of the function , when , and when vanishes on an infinite set of points of the interval .
- Prove that for .

More later, cheers,

Nalin Pithwa.

October 23, 2018 – 8:32 pm

Another set of “**easy to moderately difficult” **questions:

- The function decreases in the interval (a) (b) (c) (d) . There are more than one correct choices. Which are those?
- The function decreases in the interval (a) (b) (c) (d) . There is more than one correct choice. Which are those?
- For , satisfies the inequality: (a) (b) (c) (d) . There is more than one correct choice. Which are those?
- Suppose exists for each x and for every real number x. Then, (a) h is increasing whenever f is increasing (b) h is increasing whenever f is decreasing (c) h is decreasing whenever f is decreasing (d) nothing can be said in general. Find the correct choice(s).
- If , when , and , when . Then, (a) is increasing on (b) is continuous on (c) doesn’t exist (d) has the maximum value at . Find all the correct choice(s).
- In which interval does the function increase?
- Which is the larger of the functions and in the interval ?
- Find the set of all x for which .
- Let , if ; and, , if . If has local minimum at , then ?
- There are exactly two distinct linear functions (find them), such that they map and .

more later, cheers,

Nalin Pithwa.

October 19, 2018 – 6:33 pm

**“Easy” questions:**

Question 1:

Find the slope of the tangent to the curve represented by the curve and at the point .

Question 2:

Find the co-ordinates of the point P on the curve , the tangent at which is perpendicular to the line .

Question 3:

Find the co-ordinates of the point lying in the first quadrant on the ellipse so that the area of the triangle formed by the tangent at P and the co-ordinate axes is the smallest.

Question 4:

The function , where is

(a) increasing on

(b) decreasing on

(c) increasing on and decreasing on

(d) decreasing on and increasing on .

Fill in the correct multiple choice. Only one of the choices is correct.

Question 5:

Find the length of a longest interval in which the function is increasing.

Question 6:

Let , then is

(a) increasing on

(b) decreasing on

(c) increasing on

(d) decreasing on .

Fill in the correct choice above. Only one choice holds true.

Question 7:

Consider the following statements S and R:

S: Both and are decreasing functions in the interval .

R: If a differentiable function decreases in the interval , then its derivative also decreases in .

Which of the following is true?

(i) Both S and R are wrong.

(ii) Both S and R are correct, but R is not the correct explanation for S.

(iii) S is correct and R is the correct explanation for S.

(iv) S is correct and R is wrong.

Indicate the correct choice. Only one choice is correct.

Question 8:

For which of the following functions on , the Lagrange’s Mean Value theorem is not applicable:

(i) , when ; and , when .

(ii) , when ; and , when .

(iii)

(iv) .

Only one choice is correct. Which one?

Question 9:

How many real roots does the equation have?

Question 10:

What is the difference between the greatest and least values of the function ?

More later,

Nalin Pithwa.

August 6, 2018 – 6:55 am

July 17, 2018 – 12:19 am

**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

*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.