Calculus I, II, II, and differential equations

College Park Tutors Blog

College Park Tutors Blog

Apr
06,
2013

Math can often feel overwhelming. Sometimes, it just **looks** overwhelming, even when you actually know how to do it. This visual intimidation factor is something that breeds anxiety, induces brain farts on tests, and ultimately turns a lot of people off to math. Case in point, our calculus problem this evening:

Example: Find the first derivative of:

\(f(x)=ln\left (\frac{(3x-7)^{4}(x+1)^{2}e^{2x}}{ln(x)\sqrt{2x^2-x}} \right )\)

Hint: There is an easy way, and a hard way to approach this problem. Use the easy way.

This function, frankly, looks terrifying. Its full of exponents and natural logs and square roots, all jammed up against one another in some night-before-the-test anxiety dream. The hint provides a glimpse of comfort, if only we can figure out the "easy way." If not, we'll be doing chain rules inside product rules inside quotient rules inside chain rules all night, run out of time, and probably fail the exam. Basically, if you see a problem like this and don't know the trick, you're better...

Apr
19,
2015

The chapter on series is notorious for making calculus students want to hurl their textbooks into a fire, curl up into a fetal position, and sob uncontrollably the night before the exam (I've been there myself).

Honestly, its not that series are so difficult as much as it requires you to remember a lot of rules, how to apply them, and most importantly, in what order to apply them. And most calculus courses just don't do all that great a job of explaining this - they just hurl a bunch of theorems at you, a few tests, and tell you to sort it all out yourself. To help with this, I've created a flowchart that explains how to approach solving a mathematical series. The important things to keep in mind are:

- There is a BIG difference between finite and infinite series. Look at the problem carefully before you decide how to approach it.
- Many tests are inconclusive about whether an infinite series converges/diverges. In that case, you must try another test.
- Just because we can prove that an infinite series converges, doesn't mea...

Feb
18,
2018

## Review problem - maximizing the volume of a fish tank

Optimization, or finding the maximums or minimums of a function, is one of the first applications of the derivative you'll learn in college calculus.

In this video, we'll go over an example where we find the dimensions of a corral (animal pen) that maximizes its area, subject to a constraint on its perimeter. Other types of optimization problems that commonly come up in calculus are:

- Maximizing the volume of a box or other container
- Minimizing the cost or surface area of a container
- Minimizing the distance between a point and a curve
- Minimizing production time
- Maximizing revenue or profit

This video goes through the essential steps of identifying constrained optimization problems, setting up the equations, and using calculus to solve for the optimum points.

You're in charge of designing a custom fish tank. The tank needs to have a square bottom and an open top. You want to maximize the volume of the tank, but you can only use 192 square inches of glass...

Mar
29,
2015

Tonight, on the World's Most Extreme Values. One 2-variable function. One closed region. One shot at glory. Don't miss it!

...sorry, had to get that out of my system. The problem we're going to look at today goes like this:

Find the absolute minimum(s) and maximum(s) of the function \(f(x,y)=xe^y-x^2-e^y\) on the rectangle with vertices \((0,0)\), \((0,1)\), \((2,0)\), and \((2,1)\).

Ok, we've seen extreme value (i.e., maximum and minimum) problems like this in Calculus 1. If you don't remember the gist of this, please go back and check your notes/textbook first. Just to review, the basic idea is that we find the derivative of a function, set it equal to zero, and solve the resulting equation. Together with the points where the the function is non-differentiable, these solutions give us a set of critical points where the function might have a maximum, minimum, or inflection point.

Our example has two new issues we must confront. First of all, we have a function of two variables, so what does it mean to "...

Mar
02,
2015

Howdy folks, Alex here! I thought I'd start this blog off right, with one of the most popular (and head-spinning) problems that get thrown at my calculus students. The problem goes something like this:

Find the area of the region enclosed by the lines \(y=2x\), \(y=3x\), and \(y=2\). You must use calculus or you will not receive any credit!

Whoa. Strong words from the guy with the gradebook. Alright, well hopefully you've already seen problems that ask for the area between *two* functions. If not, break out your textbook ;). The two-function area problems are solved by integrating the difference between the "top" and "bottom" functions, like so:

But wait a minute! Our problem is giving us THREE functions, not two. How in the name of Bieber can we apply the above formula to a region enclosed by three functions?

I'll tell you how. We're going to figure out a way to **decompose**, or break down, this difficult problem into several smaller, easier problems. Learning how to break down problems, by the way, is t...

May
18,
2017

### by Jesse Salsbury

## What is the natural logarithm?

This part is optional, as students are unlikely to be tested on this material, but if you would like a better understanding of what the natural logarithm means conceptually, you may want to read this section.

Sometimes students will see \(ln(x)\) on a paper, refer to it as "el-en", but not know what it actually means. Perhaps the easiest way to understand it is to know its relation to the number, \( e\). Remember: \(e\) is just a constant, approximately equal to \(2.71828\). ** \(e^x\) ** and ** \(ln(x)\)** are inverses.

What does this mean? Think about cubing a number and taking the cube root of a number. Imagine that we start out with the number \(7\). If we cube it, we get \(343\). If we take the cube root of this new number, \(343\), we get \(7\), which is the number we started with. Similarly, imagine that we start out with the number \(125\). If we take the cube root, we get \(5\). If we cube this new number, \(5\), we get \(125\), which is the numbe...

Mar
22,
2015

Or more specifically, a second-order *linear* homogeneous differential equation with complex roots. Yeesh, its always a mouthful with diff eq. Oh and, we'll throw in an initial condition just for sharks and goggles. The problem goes like this:

Find a real-valued solution to the initial value problem \(y''+4y=0\), with \(y(0)=0\) and \(y'(0)=1\). Your solution must be real-valued or you will not receive full credit!

If you'll recall, the steps for solving a second-order homogeneous diff. eq. are as follows:

- Write down the characteristic equation.
- Find the roots of the characteristic equation.
- Use the roots to write down the two exponential basis solutions.
- Create a general solution using a linear combination of the two basis solutions.

For step 1, we simply take our differential equation and replace \(y''\) with \(r^2\), \(y'\) with \(r\), and \(y\) with 1. Easy enough:

For step 2, we solve this quadratic equation to get two roots. The roots are going to be complex numbers, but that's ok:

...
Mar
27,
2015

If you're reading this post, you've probably arrived at the point in your calculus course where they try to teach you how to do integration by partial fraction expansion. And if you're anything like I was when I first learned calculus, you're probably scratching your head and going **whadafuhhhh**? **What is the point of all this**?

Here's what you need to know. *Partial fraction expansion is not an integration technique*. It's an algebraic technique. That being said, it's useful for making certain algebraic expressions (i.e. rational expressions) easier to integrate by breaking them into smaller, simpler chunks. Sound familiar? Its that same mode of thinking that all your other calculus classes are pushing at - taking a difficult problem, decomposing it into several easier problems, and then putting the pieces back together.

Got it? Let's get down to business. Here's our problem:

Rewrite the following expression as the sum of three partial fraction terms: \[\frac{1}{(x-1)(x^2+x+1)}\]

Just to quickly review the step...

Mar
10,
2016

A lot of the "word problems" that come up in calculus seem silly and contrived, because they are. The inventory cost problem, however, is something that comes up in real-life manufacturing scenarios all the time - how can I minimize my operating costs? In fact, the problem we see here today is a simplified version of a problem I covered in a DETC conference paper that I published a few years back.

Hot Bod Jacuzzi & Spa Company is launching a new hot tub - the Neverleak Massage-o-matic DeLux. They have an exclusive deal with Gallmart to supply the retail giant with 10,000 units over the next several years. The hot tub shells are made using injection-molding, in which molten plastic is squirted into metal molds at high pressure, and then allowed to cool. Once the shell has cooled, assembly workers finish the product by attaching the hoses and motors and installing insulation.

Being a small company, Hot Bod doesn't have their own factory - they will have to rent space from the Berry Plastics Corporation. Berry Plastics charg...