In this tutorial, I will outline how to find the derivative of an inverse function using Theorem 3.17. That is, we will find the derivaitve of an inverse function without knowing the explicit definition of the inverse function....

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In this tutorial, I will demonstrate how to work with basic discrete and binomial distributions. We will find the mean and standard deviation of each type. We will also discuss how to use a TI-83 series calculator or binomial table to find specific binom...

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In this short tutorial I will demonstrate how to find the equation of the tangent line at a point on an implicitly defined function. We first will find the implicit derivative, dy/dx, and then evaluate it at the point (x,y) of interest to find the slope. ...

From Stats Math Etc.

In this tutorial, I demonstrate, using different techniques, how to find the derivative of a quotient (when the chain rule is also involved). In many cases, one can use the product rule or the quotient rule. One also must recall the properties of exponent...

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In this tutorial, we demonstrate how to create piece-wise functions that are continuous over the entire real number line. We are given the outline of functions but then have to find the particular function that establishes continuity....

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In this tutorial, I will explain several additional related rates problems from calculus 1. If you would like to watch my original lecture on related rates, check out my YouTube calculus 1 lectures playlist....

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In this lecture, I will introduce the concept of an infinite limit. An infinite limit is the scenario when a limit fails to exist due to unbounded behavior toward positive or negative infinity. We will also discuss how infinite limits dovetail with vertic...

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In this lecture, I will introduce, and demonstrate via multiple examples, the technique of integration by substitution. This technique is typically abbreviated as u-substitution. U-substitution allows us to antidifferentiate composite functions. Essential...

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In this short tutorial, I demonstrate how find upper and lower sum approximations for cos(x). I also show a quick example evaluating a definite integral and how symmetry can help us simplify our calculations....

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In this lecture, I will introduce the Fundamental Theorem of Calculus (FTC). The FTC allows us to evaluate a definite integral must faster than using Riemann Sums. As long as we can find an antiderivative, we can apply the FTC. We will also discuss the se...

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In this lecture, I will finally link the concept of integration with area. When a function is non-negative, the definite integral will yield area. We will generalize the limit definition we used in Section 5.2 - this is called a Riemann Sum. We will also ...

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In this lecture, I will introduce the concept of "area under a curve." Finding the area between a curve and the x-axis is a major topic of integral calculus. In this lecture, I will demonstrate how to find area using the limit definition. I will also disc...

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In this lecture, I will introduce the concept of the antiderivative. Antidifferentiation is the process of finding a function F(x) whose derivative is a function f(x). We will be "reversing" the process of differentiation. Please note antidifferentiation ...

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In this short tutorial, I will show how to estimate the area between the x-axis and a function using a left hand Riemann Sum. In this example, we are not finding the exact area under the curve since we are not taking the limit as n approaches infinity. He...

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In this lecture, I will demonstrate how to synthesize many of our previously studied chapter 3 and 4 techniques together in order to sketch functions by hand. I will also use algebraic techniques such as axes intercepts, symmetry, root multiplicity, and ...

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In this lecture, I will explore how to use differential calculus to optimize functions. There are no new theorems or concepts in section 4.7. However, we will need to combine lots of mathematical concepts in order to solve these application problems....

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In this lecture, I will revisit the concept of tangent lines and using them to approximate a function near a point of tangency. We will call this a tangent line approximation or linear approximation. I will also discuss the concept of a differential and i...

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In this lecture, I will discuss the concept of concavity. Concavity refers to how the curvature of a function is opening upward or downward. I will also introduce the Second Derivative Test. The Second Derivative Test, like the First Derivative Test, iden...

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In this lecture, I will revisit the limit concept. We will extend the limit concept to include limits as x is increasing or decreasing without bound. This will allow us to determine the end behavior of functions. As previous chapter 4 concepts, this will ...

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In this lecture, I will introduce the First Derivative Test. The First Derivative Test allows us to find the relative extrema of a function by first observing where a function is increasing or decreasing. As in previously discussed, relative extreme occur...

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In this lecture, I will introduce Rolle's Theorem and the Mean Value Theorem (MVT). Rolle's Theorem guarantees that at least one extrema can be found on an open interval if three prerequisite conditions are met. On the other hand, the MVT ensures that the...

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In this lecture, I will introduce the concept of extrema. Extrema are maximum or minimum values of a function - either on a specific interval or the entire domain. These can be absolute or relative. In this lecture, we will focus on finding absolute extre...

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In this lecture, I demonstrate how to solve various related rates exercises. The general strategy will be 1) find an equation to relate the desired variables, 2) differentiate with respect to t, 3) substitute in known information into the equation from st...

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In this short demonstration, I show how to find the equation of a tangent line when dealing with the inverse trig function arctan3x. We will use the usual process:
1) Find f'(x)
2) Evaluate f'(x) at the desired point
3) Use point-slope form to write the ...

From Stats Math Etc.

In this short tutorial, I show how to find the derivative of y = 9arccos(x/8). This example involves some slightly tricky simplification. The calculus is pretty straight forward. Always be careful with your algebra and arithmetic when simplifying derivati...

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In this short tutorial, I demonstrate two methods for finding the derivative of an exponential function with a variable base. I first show how to use logarithmic/implicit differentiation. Lastly, I show how to rewrite f(x) first using algebraic techniques...

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In this short tutorial, I explain how to find the equation of a tangent line of arcsec(6x) at a specific point. The process is the same as it has been in the past - we are now just dealing with inverse trigonometric functions. Thus, we need to find f'(x),...

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In this short tutorial, I demonstrate how to determine if two implicitly defined curves are orthogonal. We first need find the derivative of each equation. We then need to compare the derivatives. If each derivative is the negative reciprocal of the other...

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In this short tutorial, I will demonstrate how to find the equation of a tangent line at a point on an implicitly defined curve. Furthermore, I will show how (in this case) we can also use explicit differentiation to arrive at the same conclusion....

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In this tutorial, I will demonstrate how to find the equation of the tangent line at a point. We have done this routine many times at this point. However, this time is a little more difficult because we are dealing with an implicit equation with inverse t...

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In this lecture, I will discuss the relationship between two inverse functions' derivatives. There is a pretty cool reciprocal property between the slope of a function and the slope of its inverse. The only tricky part of this discussion is keeping domain...

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In this short tutorial, I will demonstrate how we can, in some circumstances, rewrite an implicit function in explicit form. We can then find the slope of a tangent line using the implicit derivative or the explicit derivative....

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In this lecture, I will introduce the concept of implicit differentiation. This is the technique that must be used if we are trying to differentiate an equation that cannot be written in explicit form. We will also use implicit differentiation to find der...

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In this lecture, I introduce the concept of the chain rule. The chain rule is used to differentiate composite functions. This will allow us to differentiate many more functions than we were previously capable of differentiating. I will also discuss how to...

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In this short tutorial, I will demonstrate how to use the quotient rule to find a derivative. I will also show how the same solution can be reached (in this case) using the power rule (with a little bit of algebra beforehand). Lastly, I show how to find t...

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In this lecture, I will introduce the product and quotient rules for finding derivatives. We will also show why these rules work, using the limit definition of the derivative. I will also discuss how to find the second, third, fourth, etc...derivative of ...

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In this short tutorial, we will discuss how to find the derivative of f(x) = (sinx)/x. We will also evaluate f'(x) at c = pi/4 (This is easier said than done!). We will then go beyond the original question and find the equation of the tangent line of f(x)...

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In this short tutorial, I demonstrate how to solve a couple of the homework exercises from sections 3.3 and 3.4. Specifically, we will cover a quotient rule derivative (that can also be solved using the power rule), a product rule derivative, and chain ru...

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In this lecture, I introduce several of the basic rules of differentiation. These rules will allow us to find the derivatives of constants, power functions, sums or differences of functions, basic trigonometric functions, and the exponential function. We ...

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In this lecture I introduce the concept of the derivative. I also discuss two different limit definitions of the derivative. I also demonstrate how to find the equation of a tangent line to a curve at a point. Lastly, we will also discuss how continuity a...

From Stats Math Etc.