72 min. Review of the composition of functions. Formulas and examples how to compute the derivative of composite functions.
| Notes 54 min. The formulas and examples of differentiating products and quotients.
| Notes 65 min. Explanation of the limit laws including the Squeeze Theorem.
| Notes 56 min. An explanation of continuity of functions of one variable. Classes of functions that are continuous. Summations, products, compositions of continuous functions are continuous. Other classes of continuous functions are polynomials, rational functions, exponential functions, trigonometric functions, logarithmic functions, etc.
| Notes 34 min. Secent lines converging to tangent lines. The slope of the tangent line. Instantaneous rate of change. Velocity. Simple examples illustrating the direct computation of the slope of the tangent line.
| Notes 37 min. Derivations of formulas for derivatives of ln x, ln u, ln g(x). Examples of differentiation process for composite functions that invlove logarithms.
| Notes 58 min. Introductory lecture on maxima and minima. Global absolute maxima, minima, extrema. Relative minima and maxima. Critical numbers.
75 min. Infinite limits, limits at infinity. Vertical and horizontal asymptotes.
65 min. Definition of continuity of f(x) at a number x=a. A variety of pictorial examples illustrating various types of discontiuities. Continuity from the right, continuity from the left.
71 min. Applications of derivatives.
51 min. Implicit differentiation techniques. Illustrations based on examples.
| Notes 74 min.
53 min. We outline the key problems in calculus: the area problem and the tangent problem. Both of these are defined using limits. The limit of a function f(x) as x approaches a number a gives information about he tendency of the values of f(x) for x near a.
52 min. Limits from the left and right. Limit Laws and examples illustrating how to use them.
55 min. Various ways how a function may fail to be continuous. Continuity from the right, and from the left. Examples. Classes of functions that are continuous
57 min. Limits as x \to \infty. Divergence of f(x) as x \to a from the left and right. Vertical and horizontal asymptotes. Examples how to calculate limits
48 min. Introduction to rates of change in mathematics and applications.
67 min. Derivative at a number x=a and the derivative function.
34 min. Basic differentiation rules. The derivation of \frac{d}{dx} (e^x) =e^x.
31 min. Examples using the Product Rule: (f g)' = f' g + f g' and Quotient Rule \left (\frac{f}{g} \right )' = \frac{f' g - f g'}{g^2}.
42 min. A quick review of trigonometric functions. Sines, Cosines, Tangents, Secants. Summation formula for sines and cosines. Trigonometric limits, list of trigonometric derivatives. Examples.
64 min. The explanation of the Chain Rule: (f \circ g)' = f'(g(x)) g'(x) and numerous examples how to use it.
55 min. The Chain Rule applied to equations involving x and y. Examples.
44 min. We show that \frac{d}{dx} ( \log_a x ) = \frac{1}{x \ln a}. Then we illustrate examples where we take the logarithm of both sides of an implicit equation before finding the unknown derivative. Logarithmic differentiation allows us to find derivatives of functions such as this one f(x) = x^x.
20 min. An explanation of how to solve related rates problems based on a simple example. The volume of the ballon is increasing at a rate of 100~cm^3/s. How fast is the radius of the ballon increasing when it is equal to 20~cm?
59 min. Introduction to extrema. Absolute maxima, minima, local maxima, minima, critical numbers. Closed Interval Method.
67 min. Rolle's Theorem. The Mean Value Theorem. Examples of functions that do not satisfy the Mean Value Theorem. Tests for increasing and decreasing. Tests for concavity.
66 min. Not all critical numbers give rises to local max or min. To ensure that a critical number is a local min or max, the derivative must change its sign. An inflection point is a point on the graph of y=f(x) at which the function changes its convavity.
67 min. Explanation and examples of limits of the indeterminate forms \frac{0}{0}, \frac{\pm \infty}{\pm \infty}, \infty - \infty, 0^0.
57 min. Graph sketching techniques based on the example: f(x) = \frac{x^2}{ \sqrt{x+1}}. When evaluating a limit, the instructor makes a mistake. Can you spot it before he corrects it?
65 min. Examples of optimization problems. Steps in solving these. The key in optimization problems is to determine a function f(x) that needs to be minimized or maximized and determine its domain. If the domain is a closed interval and then a Closed Interval Method can be applied to find max and min. If the domain is not a closed interval, then sometimes the First Derivative Test for absolute extrema can be applied.
56 min. Finding functions F(x) whose derivatives are known.
53 min. \begin{pspicture}[plotpoints=200](-0.25,-.02)(5.2,4.2) \psStep[algebraic,StepType=lower,fillstyle=solid,fillcolor=yellow] (1.00,4){6}{ (1.6*sin(x))^2+.7} \psplot[algebraic,linecolor=red,labelFontSize=\footnotesize]{0.001}{5}{(1.6*sin(x))^2+.7} \psaxes[labels=all,labelcolor=red]{>}(0,0)(-0.1,-0.75)(4.5,4.5) \psplot[algebraic,linecolor=red,labelFontSize=\footnotesize]{0.001}{5}{(1.6*sin(x))^2+.7} \uput[90](1.95,3.3){$\Delta x = \frac{b-a}{n}$} \uput[90](.59,-0.75){$a=$} \uput[90](3.59,-0.75){$b=$} \uput[90](1.76,-0.1){\scriptsize $\Delta x$} \uput[90](2.26,-0.1){\scriptsize $\Delta x$} \psline(4,0)(4,2.3) \end{pspicture} A fundamental problem in Calculus is to find areas bounded by curves. Here we start by estimating a simple case. In order to estimate the area under the graph y= f(x) above the X-axis and between x=a and x=b, we divide the interval [a,b] into n pieces each of the same length \Delta x = \frac{b-a}{n}. In each of those intervals we pick a number x_i^*. We then sum the areas of the rectangles of width \Delta x and height f(x_i^*). Elementary Riemann Sums with a small number of rectangles. Introduction to definite integral. Left endpoint rectangles, right endpoint rectangles, midpoint rectangles.
49 min. The definition of the area under the graph of y=f(x) (of a positive function) above the X-axis and between x=a and x=b. \int_a^b f(x) dx = \lim_{n \to \infty} \sum_{i =1}^n f(x_i^*) \Delta x. An Example how to use limits to calculate the area under a linear function. The sigma notation.
45 min. Introduction to the Fundamental Theorem of Calculus. We define the area function g(x) = \int_a^x f(t) dt. Then g'(x) = f(x) or \frac{d}{dx} \int_a^x f(t) dt = f(x). The lecture ends with an open ended question.
27 min. The evaluation part of the Fundamental Theorem of Calculus and its derivation. Basic examples that illustrate how to evaluate a definite integral using antiderivatives.
\begin{pspicture}[plotpoints=200](-0.25,-.02)(2.2,3) \psStep[algebraic,StepType=lower,fillstyle=solid,fillcolor=yellow,linecolor=yellow] (0.4,1.6){100}{1+sin(x)} \rput(1.2,2.3){$y = f(t)$} \psaxes{>}(0,0)(-0.1,-0.75)(2.5,3.5) \psplot[algebraic,linecolor=red,labelFontSize=\footnotesize] {0.1}{2}{1+sin(x)} \uput[90](0.97,.7){\scriptsize $\int_a^b f(t) dt$} \uput[90](0.46,-.37){\scriptsize $a$} \uput[90](1.66,-.37){\scriptsize $b$} \end{pspicture}
In the derivation we start by recalling that \frac{d}{dx} \int_a^x f(t) dt = f(x), hence G(x) = \int_a^x f(t) dt is an antiderivative of f(x). In particular, G(b) = \int_a^b f(t) dt. Hence, we can compute \int_a^b f(t) dt if we know the antiderivative G(x).

To calculate it, suppose that we know some antiderivative F(x) of f(x). Any two antiderivatives differ by a constant, G(x)= F(x) -C. Since G(a) =\int_a^a f(t) dt = 0, C= F(a). Thus, G(x) = F(x)-F(0). We then obtain Let F(x) be any antiderivative of a continuous function f:[a,b] \to \R. Then \bf \int_a^b f(t) dt = F(b)-F(a).

62 min. We summarize both parts of the Fundamental Theorem of Calculus. We then compute several elementary integrals, two of which have absolute values. We conclude the lecture with an example where we investigate the area function. This is the last lecture in Calculus 1 series.
33 min. f(x) = \frac{x}{x^2-4}. We start with the domain, then we look for symmetries, asymptotes, intervals of increase and decrease, the concavities.
46 min. An introduction to average and instantaneous rates of change in calculus, sciences, and finance.
| Notes 13 min. How to submit online calculus homework in Spring 2009.
45 min. We illustrate how to find the derivatives at numbers using the definition of the derivative. We show examples how to sketch the graph of $f'(x)$ based on the graph of f(x).
25 min. Using the definition of of the derivative, we illustrate and derive basic differentiation formulas. We define the number $e$ and prove that $(e^x)'=e^x$.
26 min. A Calculus 1 basic introduction to differential equations. Exponential growth and decay equation. How to verify if a function is a solution of a differential equation.
48 min. The unit circle. Definitions of the sine, cosine, tangent functions. Also defined are cotangent, secant, and cosecant. Basic properties of sine and cosine. The trigonometric identity $cos^2 \theta + \sin^2 \theta =1$. Sine and cosine of the summation. Double angle formula for sine. Special angles, $pi/4$ and $pi/3$.
71 min. Review of the definitions of sine and cosine. The derivation of $\frac{d}{dx} \sin x = \cos x$. Trigonometric limits. Other derivatives of trigonometric functions.
53 min. Learning cycle in the online calculus class. The area problem. An example how to rearrange pies and obtain a formula $A=\pi r^2$. The tangent problem. The limit as a common approach in both fundamental problems in calculus.