Practical Calculus Applications

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How to Use Differentiation to Calculate the Maximum Volume of a Box

One of the most practical uses of differentiation is finding the maximum or minimum value of a real-world function. In the following example, you calculate the maximum volume of a box that has no top and [more…]

How to Use Differentiation to Calculate the Maximum Area of a Corral

Finding the maximum or minimum value of a real-world function is one of the most practical uses of differentiation. For example, you might need to find the maximum area of a corral, given a certain length [more…]

Related Rates: the Expanding Balloon Problem

Say you’re filling up your swimming pool and you know how fast water is coming out of your hose, and you want to calculate how fast the water level in the pool is rising. You know one rate [more…]

Related Rates: the Trough of Swill Problem

Say you’re filling up your swimming pool and you know how fast water is coming out of your hose, and you want to calculate how fast the water level in the pool is rising. You know one rate [more…]

Related Rates: Two Cars at a Crossroads

Say you’re filling up your swimming pool and you know how fast water is coming out of your hose, and you want to calculate how fast the water level in the pool is rising. You know one rate [more…]

How to Determine Marginal Cost, Marginal Revenue, and Marginal Profit in Economics

Marginal cost, marginal revenue, and marginal profit all involve how much a function goes up (or down) as you go over 1 to the right — this is very similar to the way linear approximation works. [more…]

How to Do a Related Rate Problem Involving a Moving Baseball

You can use calculus to determine a rate that’s related to the speed of a moving object. For example, say a pitcher delivers a fastball, which the batter pops up — it goes straight up above home plate. [more…]

How to Calculate a Sequence's Terms by Using the Sequence Expression

If you're given a few terms of a sequence, you can often use these terms to find the general formula for the sequence. If you're given the general formula [more…]

How to Find the nth Term Using the First Terms of an Arithmetic Sequence

If you know the first few terms of an arithmetic sequence, you can write a general expression for the sequence to find the nth term. To write the general expression, you must look for a pattern in the [more…]

How to Recognize Recursive Arithmetic Sequences

A recursive sequence is an arithmetic sequence in which each term depends on the term(s) before it; the Fibonacci sequence is a well-known example. When your pre-calculus teacher asks you to find any term [more…]

How to Find the General Formula for the nth Term of an Arithmetic Sequence Using Any Two Terms

At some point, your pre-calculus teacher will ask you to find the general formula for the nth term of an arithmetic sequence without knowing the first term or the common difference. In this case, you will [more…]

How to Identify a Term in a Geometric Sequence When You Know Consecutive Terms

If your pre-calculus teacher gives you consecutive terms in a geometric sequence and asks you to identify another term in the sequence, the steps you will follow to find this term are remarkably similar [more…]

How to Use Consecutive Terms to Find Another in an Arithmetic Sequence

If your pre-calculus teacher gives you two consecutive terms of an arithmetic sequence and asks you to find another, you can use a general formula to find the common difference between these terms. For [more…]

How to Identify a Term in a Geometric Sequence When You Know Two Nonconsecutive Terms

If your pre-calculus teacher gives you any two nonconsecutive terms of a geometric sequence, you can find the general formula of the sequence as well as any specified term. For example, if the 5th term [more…]

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