Calculus with Analytic Geometry by Joe Repka

Legend says that an ancient king was so pleased with the new game of chess that he offered a huge reward to its inventor. The wise man asked the king to put one grain of wheat on the first square of the chessboard, two on the second, four on the third, and so on, doubling each time. 

Calculate the amount of wheat on the sixty-fourth square, assuming a grain of wheat weighs 50mg and bearing in mind that the annual worldwide wheat harvest is about 500 metric tonnes. 
Incidentally, the phrase "check mate" is apparently a corruption of 'shah matt" which in old Persian means "the king is dead".
(Take this problem as an exercise. You'll be astonished by the results. Btw, don't forget to use common units of measure)

This was an introduction to the chapter Logarithms and Exponential Functions in the book Calculus with Analytic Geometry by Joe Repka. This is an awesome book for Engineering Mathematics with detailed explanations. Numerical examples are used to illustrate concepts (unlike the books engineering students are made to study from in which the sole aim is to arrive at a solution without any concrete foundation). If you are interested in studying calculus this is a great book to start off with.

Some of the interesting intros I found in the book are as follows :

Limits and Continuity

Aristotle declared that heavy objects fall faster than light ones, in proportion to their weight, and for centuries the world believed him.
Finally Galileo showed otherwise, reputedly by dropping a cannonball and a musket ball from the tower of Pisa. He is often regarded as the father of experimental science (Aristotle purported to have conducted similar experiments, but presumably never did). 
Galileo's work lead to Newton's study of gravitation, which in turn led to the development of calculus.

Derivatives

Newton studied gravity, reputedly after watching an apple fall from a tree. Combining this work with his recently developed techniques of calculus, he was able to show how the laws of gravity explained the motions of the planets. This had been a great mystery, and his solution established both Newton's reputation and the importance of calculus.

Integration

To find the area of a circle, we tried approximating by a number of triangles and adding up their areas. Then we took a limit as the number of triangles goes to infinity.
To find the area of a region enclosed by a curve, we approximate the region by a number of rectangles and add up their areas. If we use narrower rectangles, the approximation should be more accurate. The exact area is found by taking  a limit as the rectangles get narrower and narrower.

Series

A hare trying to pass a tortoise must first catch up half the distance between them. Then he must catch up half the remaining distance. Then he must catch up half the remaining distance, and so  on.
The argument known as Zeno's paradox, suggests that the hare can never overtake the tortoise.

Vector Calculus

The Mobius strip challenges our intuitive ideas about surfaces. Take a strip of paper, make a single twist in it and tape it into a loop. With a pencil draw a line down the center of the strip without crossing the edge of the paper. You will find that the pencil has returned to the back of the paper. For this reason people say that the Mobius strip has only one side. 


If you have performed this experiment, you might like to try another by taking a pair of scissors and cutting along the line down the middle of the strip of paper...


Current Song:
Wish You Were Here by Pink Floyd