Differential Calculus
- Functions and Graphing
- Limits and Rates of Change
- Derivatives
- Finite Sequence and Series
Differential Calculus
Differential Calculus
Limits of Radical Functions Pt. 2
In this pencast I provide additional examples of finding the limit of a radical function, and in particular, using the conjugate method.
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Limits of Radical Functions Pt. 1
Radical functions are difficult to account for in an algebraic expression. They especially pose problems when you are trying to find limits. In this pencast I discuss how to account for them by multiplying by their conjugates.
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More Limits
Additional examples of evaluating the limit of a function.
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Chain Rule Examples 2
In this pencast continue the work of a previous pencast by providing a few examples of using the chain rule.
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Chain Rule Examples 1
In this pencast I use the chain rule to find the derivative of several equations.
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Parametric Piecewise Functions
In a recent pencast, I discuss the utility of piecewise functions. In this one I discuss how to manipulate a function by using parameters — variables that I can pick to make a function behave in a particular, useful way.
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Piecewise Limits
You may wonder what piecewise functions are all about. Why are they useful? In this pencast I answer these questions and find the limit of a piecewise function.
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Trigonometric Limits
You may think you have the topic of limits covered… But can you handle trigonometric limits? In this pencast I find the limit of a trigonometric function. You may think that they are periodic, and therefore don’t have any limit value? Not necessarily…
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Finding Limits Involving Radicals Part 2
In this pencast I continue my discussion of finding the limit of a function that includes radicals. I also discussion a realistic example of the subject.
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Finding Difficult Limits
Limits are a subject that can easily confuse a student. In this pencast I solve a problem involving a particularly difficult limit equation
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Find Solutions to Sine Functions
In this problem I find the derivative of more complicated trigonometric functions and use this information to solve a unique problem.
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Parameters and Tangent Lines
In this problem I introduce the concept of a parameter within a function. Technically, a parameter is a variable, but for some purposes they work as an input “switch” or control value.
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Trigonometric Composite Functions
Composite functions can be difficult to understand. Things get more complex when you throw trigonometric functions into the equation-situation (ha.) In this pencast I find the derivative of such a function.
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Chain Rule Derivation
The chain rule and power rule outline two different methods to accomplish the same goal: derivation. In this problem I use both methods to solve a problem and discuss the usefulness of each.
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Driving with Derivatives
Driving, position, velocity and acceleration are all related topics. In this pencast I relate these concepts to each other, providing a unique physical example of the concept of a second derivative.
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Find Extrema and Classify Function
In this problem, I use derivatives to find the local maximums and minimums of a function and determine the intervals on which the function is concave up or concave down.
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Finding Inflection Points
Trigonometric functions have plenty of inflection points, local maximums and local minimums. Why? Because they’re periodic — they repeat. Selecting a particular interval gives you the opportunity to find a limited number of inflection points.
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Cylinder Linear Approximation
Linear approximations aren’t very intuitive when you’re only looking at symbols on a page. In this problem, I make it easier to understand how and why linear approximations are useful, and make it easier to digest by using a physical example — a cylinder.
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Linear Approximation
A function’s graph may have maximums, minimums, spikes and strange looking things. These characteristics can make it difficult to calculate the derivative of the function. Here’s where linear approximation can come to the rescue: Functions can be approximated by simple lines! But beware — this is only useful for under certain [...]
Changes in the Earth’s Volume
Measurement is an inherently uncertain subject. Whenever you measure something, there is some error involved — hopefully you know what it is. If you know what the Earth’s radius is and what the error in your measurement might be, you can find the volume of the Earth. A change in the earth’s [...]
Fuel Efficiency of a Car
How fast should you drive a car when you’re trying to save money? How about making money? In this problem, I find the optimum speed a chauffeur should drive considering his car’s gas milage and his hourly rate. Optimization problems make calculus more interesting because they are real world problems, and your [...]
Evaluate the Limit of a Function… as it Approaches Infinity
In a previous pencast, I evaluate the limit of a function as it approaches a particular point. But what about evaluating the limit of a function as you move along the x-axis toward infinity? Holes exist as points on a function’s graph. Sometimes you don’t need to find out where those holes [...]
Absolute Value Functions, Holes, and Limits
With absolute value functions, it’s always important to keep in mind what domains apply to your function. With this equation we’re presented with the pesky problem of dividing by zero, and the consequences of having holes in the graph. Don’t worry; because with limits, everything starts making sense. But beware, sometimes a [...]
Evaluating the Limit of a Rational Function, at a Point
Always remember that you can’t divide by zero. Up until now, stumbling upon a situation where you divide by zero leaves you with a mysterious problem that you don’t know how to solve. However, if you understand what limits are, you can figure out what’s going on when you divide by zero. [...]
Graphing a Function with Holes
Some functions are full of holes when you graph them. What are holes, you might ask? Well, they’re not the kind that will cause a leak. Knowing what holes are, and how they relate to limits can be problematic, but the basic idea is very intuitive and helpful.
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