A few days ago was I wrote about Linear Approximations. These are used, as the name implies, to approximate the value of a function, usually a complicated function, by way of using a much simpler linear function, as long as \(\boldsymbol{\Delta x}\) (the change in x) is small. Along with ...
Linear Approximation – What It is with ExamplesLinear Approximation – What It is with Examples
Smooth curves appear straighter on smaller scales and it is the basis of many important mathematical ideas, one of which is linear approximation. ...
The Geometric Interpretation of the Derivative and the IntegralThe Geometric Interpretation of the Derivative and the Integral
I have been trying to learn Calculus on and off for several years. In the last 3 months I re-read about Limits, Differentiation and Applications of Differentiation and finally, reached the point where I am starting to learn about Integration. It occurred to me that it might be interesting to ...
Finding the Minimum Distance – an Optimization ProblemFinding the Minimum Distance – an Optimization Problem
Every Algebra student knows the distance formula; that is, how to calculate the distance between two points on the Cartesian coordinate plane: \(\boldsymbol{d = \sqrt{\left(x_{2} – x_{1}\right)^{2} + \left(y_{2} – y_{1}\right)^{2}}}\). This week’s blogpost is about finding the distance between a specific point and the closest point(s) in a function. ...
Second Derivative Test and Inflection PointsSecond Derivative Test and Inflection Points
One of the many interesting things I’ve come across while learning calculus is the concept of concavity. A simple example of a concave function is a parabola. The left graph below shows a concave up function: \(\boldsymbol{y = x^{2}}\), while the right graph shows a concave down function: \(\boldsymbol{y = ...
Finding the Cheapest Route, an Optimization ProblemFinding the Cheapest Route, an Optimization Problem
In this post we are going to look at a function and find its minimum value analytically. Instead of just a dry exercise, it is usually more fun to tackle a word problem. All calculus textbooks have several versions of this very same problem. The problem: An island is 3.5 ...
Using Change of VariableUsing Change of Variable
I have used change of variable while trying to solve quadratic equations. Recently, I came across some very clever uses of this technique. Some of the examples below served me as a reminder that it is OK to arbitrarily change the value of one side of an equation… as long ...
First Derivative Test – Example 2First Derivative Test – Example 2
Recently I wrote about using the First Derivative (FD) Test to find the intervals where a function is increasing or decreasing. With the FD Test, we can also find the exact value where the function is highest or lowest in an interval (called absolute maxima) or, where it is higher ...
Angle of Parabolic FlightAngle of Parabolic Flight
Although there’s an element of Calculus in this week’s post, and even more so, of Trigonometry, I am including it in the Algebra category because that is the more interesting aspect of this problem. The following equation is the path of a projectile propelled at an angle \(\boldsymbol\theta\), where: \[y ...
First Derivative TestFirst Derivative Test
Sometimes it is easy to see exactly where a function has its highest or lowest values just by looking at its graph. Most often, though, the answer requires analytical work. I am starting this new Blog with an example on how to find analytically on what interval(s) a function is increasing ...