Calculus Examples and Exercises

One of the very first things taught in a 1st year Calculus class is the concept of the ‘Limit’. One (simplistic) way of describing it, is by saying that the Limit of the function \(\boldsymbol{f\left(x\right)}\) is what the function ‘wants to be’ as \(\boldsymbol{x}\) approaches a specific value or approaches ...

After learning the rules (power rule, product and quotient rules, chain rule, etc.) and doing numerous exercises, a student realizes that the mechanics of Differential Calculus are not that difficult. Actually, they are almost boring, if it wasn’t for that one rule learned after all the other rules: implicit differentiation. ...

I am currently learning about this very powerful calculus tool. This tool is of great value in simplifying some functions prior to differentiation. I will try to explain simply in my own words. What are logarithms? Logarithms were invented by John Napier (1550-1617) for the purpose of simplifying calculation; basically ...

The following Algebra problem is taken from one of the American Mathematics Competitions (AMC 10) Preparation books. It is notable that this is one of many books helping 10th grade students get ready for math olympics competition. The problem is, in my estimation, of medium difficulty. It is really an ...

I have been concentrating on vectors lately because I need to obtain a certain level of proficiency with them in order to study college-level physics. The physics textbook I am using gets heavy on vectors right on chapter one. So, here’s another problem using vectors: a 5,000-pound load is lifted ...

Today I want to show an exercise demonstrating the use of vectors that is so complicated that a graphing utility is required to solve it, and yet, the underlying principle is simple once the basic nature of the problem is understood. Two cranes are lifting an object that weighs 20,240 ...

This week’s post has something to do with Physics (position, velocity and acceleration) but it is mostly about good old plain Algebra and about reading the problem carefully. For starters, since this takes place here on Earth, the acceleration of an object due to gravity is \(\mathbf{-32}\) feet per second ...

Vectors are used all the time in Physics and I am still only learning how to use them. I will try to provide a good example of their usefulness. Suppose you spent a nice afternoon fishing and now you’re getting ready to go back home. A force of 600 pounds ...

One of the first things you learn when starting college-level Physics is vectors; right in the very first chapter. Vectors are also taught at some level in some Algebra and Trigonometry textbooks. That’s where I’ve reached to in order to get a more solid understanding of them. Let’s try this ...

I just came across a calculus word problem about integration; specifically, the area between two curves. Two runners, starting at the same location, run along a straight road for 1 hour. The velocity of one runner is \(\boldsymbol{v_1\left(t\right) = 7t}\) and the velocity of the other runner is \(\boldsymbol{v_2\left(t\right) = 10\sqrt{t}}\). ...