Course Information

Course

Subject:	Mathematics
Topic:	Differential Calculus
Views:	72,945

Educator

Name:	Math Doctor Bob (Robert Donley)
Type:	Individual

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Calculus with Dr. Bob I: Limits and Derivatives

Video Lectures

Displaying all 82 video lectures.

Lecture 1 Play Video	Limits 1a - Definition and Basic Concepts Calculus: We define the notion of a limit and interpret it both numerically and pictorially. We compute the examples (a) lim_{x_1} (x^2-1)/(x-1) and (b) lim_{x_2} (x^2-6x+8)/(x^2-5x+6). We also note an important trick for multiple choice exams.
Lecture 2 Play Video	Limits 1b - Delta-Epsilon Formulation Calculus: We present the delta-epsilon definition of a limit, explain the various parts pictorially, and show how to choose delta when presented with a given epsilon. We show that delta = .1 works for f(x) = x^2 at x_0 =2 when epsilon = 1/2.
Lecture 3 Play Video	Limits 1c - Limit Failure Calculus: We consider three common cases for limit failure: vertical asymptotes, jump discontinuities, and oscillation. Functions considered (all at x=0) are 1/x, f(x) = sin(1/x), and f(x) = {x if x less than 0, x+1 if x greater than 0}.
Lecture 4 Play Video	Limits 1d - Polynomial and Rational Functions Calculus: We approach limits from a more abstract perspective. Using the limits for constants and f(x) = x, we build up the evaluation rule for polynomial and rational functions. General rules for limits algebra are given.
Lecture 5 Play Video	Limits 1e - Compositions and Squeeze Theorem Calculus: More limit rules are given. A general rule is given for limits of x^s (s any real number), and a rule is given for compositions. The Squeeze Theorem is stated and applied to the limits at 0 of x^2 sin(x) and \|x\| cos(1/x).
Lecture 6 Play Video	Limits 1f - Trigonometric Functions Calculus: Limit rules for trig functions are given. We also consider the special trig limits at x=0: lim sin(x)/x = 1, and lim (1-cos(x))/x = 0.
Lecture 7 Play Video	Examples of Limits Calculus: Examples of limits: (a) lim(2) (x^2-4x+4)(x^2-2x+1), (b) lim(2) (x^2-4x+4)/(x^2-5x+6), (c) lim(2) (x^2-2x+1)/(x^2-5x+6), (d) lim(2) (x^2-4)/(sqrt(x^2+5) - 3), (e) lim(pi/3) tan(x), (f) lim(0) (1-cos(x)/x^2)
Lecture 8 Play Video	Continuity 1a - Definition and Basic Concepts Calculus: We define the notion of continuity at a point and on an interval for a function f(x). We consider the different types of discontinuities. Also given are definitions for one-sided limits and continuity from one side.
Lecture 9 Play Video	Continuity 1b - Polynomial/Rational Functions and The Extreme Value Theorem Calculus: We note that polynomials and rational functions are continuous where defined. The Extreme Value Theroem and Intermediate Value Theorem are also given with examples.
Lecture 10 Play Video	Examples of Continuity Calculus: Find all points where the function is continuous. If a discontinuity is removable, find the best fitting point.
Lecture 11 Play Video	Fast Solution of Inequality Using Continuity Calculus: We use continuity to solve the inequality \|x-2\| gt 2x-8. We check our work by sketching the graphs and by solving with the definition of absolute value.
Lecture 12 Play Video	Bisection Method 1 Calculus: As an application of the Intermediate Value Theorem, we present the Bisection Method for approximating a zero of a continuous function on a closed interval. As an example, we consider the zero of f(x) = x^2-2 in [1,2] and find a bound for the error in our estimate.
Lecture 13 Play Video	Bisection Method 2 Calculus: As an application of the Intermediate Value Theorem, we use the Bisection Method to estimate the point x where cos(x) = sqrt(3) sin(x) on the interval [0, pi/2].
Lecture 14 Play Video	Vertical Asymptotes 1a Calculus: With the notion of one-sided limits, we consider the behavior at vertical asymptotes and define limits with values at +/- infinity. A rule for determining such limits of a rational function is given.
Lecture 15 Play Video	Vertical Asymptotes 1b Calculus: We give another example of how to determine vertical asymptotes for a rational function. We also consider the example f(x) = sin(x)/sin(2x).
Lecture 16 Play Video	Definition of Tangent Line Calculus: To motivate derivatives, we compare approximations using a best fitting point versus a best fitting line. The tangent line is defined.
Lecture 17 Play Video	Example of Tangent Line Calculus: We find the equation of the tangent line to the graph of f(x) =sqrt(1+x) at x0 = 3, and use the line to approximate sqrt(4.2).
Lecture 18 Play Video	Definition of Derivative Calculus: For a function f(x), we define the derivative f'(x) as the slope of the tangent line at x. Examples are given, and we show that differentiability implies continuity.
Lecture 19 Play Video	Power Rule for Derivatives Calculus: We show that the derivative of f(x) = x^n is f'(x) = nx^{n-1}. We apply the rule to obtain the tangent line to f(x) = x^5 at x=1, and use it to estimate (.9)^5.
Lecture 20 Play Video	Tangent Line to x^2-4x Calculus: Find all x where the tangent line to f(x) = x^2-4x has slope 0, 6, -6.
Lecture 21 Play Video	Horizontal Tangent Lines to a Polynomial Calculus: Find all horizontal tangent lines for the function f(x) = 3x^4 - 32x^3 + 72x^2.
Lecture 22 Play Video	Derivative of sin(x) and cos(x) Calculus: We find the derivatives of sine and cosine. Tools required are the trig identities for the sine and cosine of a sum, and the special trig limits. Applications include a tangent line approximation for sin(1) and finding the horizontal tangent line to the graph of cos(x).
Lecture 23 Play Video	Tangent Lines to sin(x) Calculus: Find all x such that the tangent line to f(x) = sin(x) has slope equal to sqrt(3)/2 and -sqrt(2)/2.
Lecture 24 Play Video	Motion in a Line Calculus: We define motion in a line by a position function s(t). Notions of average and instantaneous velocity are given. As an example, we consider three problem concerning the position of a rocket in motion.
Lecture 25 Play Video	The Product Rule Calculus: The Product Rule for derivatives is presented and applied to several examples. (a) Derivative of f(x) = (x^2 - 1)(x-2), (b) We rederive the power rule for the derivative of x^n if n is a positive integer, (c) f(x) = sin(2x).
Lecture 26 Play Video	General Product Rule Calculus: We explain the product rule when there is more than two functions in the product. Two concrete examples are given: (a) f(x) = x^2 (2x+1) cos(x), and (b) f(x) = x^{1/2} (x^2-1) sin(x).
Lecture 27 Play Video	Power Rule for Rational Exponents Calculus: Using the product rule, we find the power rule for the derivative of f(x) = x^{m/n} where m and n are positive integers. As an application, we find the tangent line to f(x) = x^{1/3} + x^{2/3} at x=8 and use it to approximate f(8.1).
Lecture 28 Play Video	The Quotient Rule Calculus: We present the Quotient Rule for derivatives, memory trick included. Examples include (a) the derivative of f(x) = (x^3-x^2)/x, (b) the derivative of f(x) = (x^{1/3} - 1)/(x^{1/3} + 1), and (c) the Power Rule for the derivative of x^{-m/n} where m, n are positive integers.
Lecture 29 Play Video	Trig Derivatives Calculus: We give a scheme for the derivative of the trig functions. We show how to obtain the derivative of tangent and cosecant using the quotient rule. As an application, we find the tangent line to f(x) = cot(x) at x = pi/3, and use the line to approximate cot(1).
Lecture 30 Play Video	Examples of Trig Derivatives Calculus: We compute the derivatives of (a) f(x) = x^2 tan(x), (b) f(x) = csc(x)/(1+x), and (c) f(x) = cos(x)/(1+sin(x)).
Lecture 31 Play Video	Tangent Lines for sec(x) Calculus: Find all horizontal tangent lines for f(x)= sec(x) in [0, 2\pi]. Then find all tangent lines to sec(x) with slope = sqrt(2) in [0, 2\pi].
Lecture 32 Play Video	Tangent Lines for cot(x) Calculus: Find all horizontal tangent lines to f(x) = cot(x). Then find all points where the tangent line has slope 1 and -1.
Lecture 33 Play Video	The Chain Rule Calculus: The Chain Rule is given for the derivative of a composition f(g(x)). We show special cases of the power rule and the composition of two lines. A brief proof is also given.
Lecture 34 Play Video	Example of Chain Rule 1 - Basic Examples Calculus: Use the chain rule to compute the derivatives of: (a) f(x) = (x^3 - x^2)^{1/2}, (b) f(x) = ((x-4)/(x-2))^10, and (c) f(x) = 3/(tan(x^3)).
Lecture 35 Play Video	Example of Chain Rule 2 - Approximation with Tangent Line Calculus: Find the equation of the tangent line to the function f(x) = ((x-4)/(x-2))^3 at x= 3. Use the tangent line to approximate f(3.01).
Lecture 36 Play Video	Example of Chain Rule 3 - Trig Functions Calculus: Find the derivative of the following trig functions: (a) f(x) = sin^4(x) + cos(4x), (b) f(x) = sec(x^2 + x + 1), and (c) f(x) = sin(cos(x)).
Lecture 37 Play Video	Example of Chain Rule 4 - Triple Chain Rule Calculus: Using a triple chain rule, calculate the derivative of f(x) = sin((1+x^2)^{1/2}).
Lecture 38 Play Video	Higher Order Derivatives Calculus: We define the higher order derivatives of a function f(x). We note alternative notations, and describe the physical significance of the second derivative as acceleration for motion in a line.
Lecture 39 Play Video	Graphs and Higher Order Derivatives Calculus: For the function f(x) = 2x^3 - 3x^2, we consider the connection between the graphs of f(x), f'(x), and f"(x). The linking idea is the slope of the tangent line.
Lecture 40 Play Video	Implicit Differentiation 1 - Definition and Basic Concepts Calculus: We give a procedure for finding tangent lines to graph of equations that are not functions. A checklist is given for using implicit differentiation to obtain y'. An example of a tangent line to the unit circle is given.
Lecture 41 Play Video	Implicit Differentiation 2 - Basic Example Calculus: Using implicit differentiation, find y' for the equation xy + (1+y)^{1/2} = y + 2 at the point (x,y) = (1, 3).
Lecture 42 Play Video	Implicit Differentiation 3 - Approximation with Tangent Line Calculus: Find the tangent line for the equation xy^{1/2} = 1+x at the point (1,4). Then use the tangent line to approximate the y-value of the point on the graph near (1,4) at x=1.1.
Lecture 43 Play Video	Implicit Differentiation 4 - Example with Trig Functions Calculus: Find y' for the equation xcos(y) + sin(xy) = 1 at the point (1, 2pi). This is an application of implicit differentiation.
Lecture 44 Play Video	Implicit Differentiation 5 - Higher Derivatives Calculus: Find the second derivative y" for the equation xy = 1 + y using implicit differentiation.
Lecture 45 Play Video	Related Rates Calculus: We give a checklist for solving related rates problems. Example: a stone falls into a pond, creating a ripple. When the radius of the ripple is 2 inches, the radius is increasing by 10 in/sec. How fast is the area increasing at that time?
Lecture 46 Play Video	Example of Related Rates 1 Calculus: A 10-ft. ladder leans on a wall. It slides down the wall. When the top of the ladder is 8 ft. high, the top of the ladder is falling at a rate of 20 ft/sec. How fast is the base of the ladder moving away from the wall?
Lecture 47 Play Video	Example of Related Rates 2 Calculus: A right circular cone (point down) has base radius 100 ft and height 100 ft. Water drains form the tip of the cone. When the water level is 10 ft, water drains at a rate of 2 ft/sec. How fast is the water level decreasing?
Lecture 48 Play Video	Extreme Value Theorem Using Critical Points Calculus: The Extreme Value Theorem for a continuous function f(x) on a closed interval [a, b] is given. Relative maximum and minimum values are defined, and a procedure is given for finding maximums and minimums. Examples given are f(x) = x^2 - 4x on the interval [-1, 3], and f(x) = 2 - x^{1/3} on [-1, 8].
Lecture 49 Play Video	Example of Extreme Value Theorem 1 Calculus: We apply the method for finding maximums and minimums of a continuous function on a closed interval to the function f(x) = 1/(1+sin(x)) on the interval [0, pi].
Lecture 50 Play Video	Example of Extreme Value Theorem 2 Calculus: Find the maximum and minimum values of f(x) = x^{4/3} -2 x^{2/3} - 3 on the interval [-8, 8].
Lecture 51 Play Video	Example of Extreme Value Theorem 3 Calculus: Find the maximum and minimum values of f(x) = \|x^2 - 9\| on the closed interval [-1, 4].
Lecture 52 Play Video	Rolle's Theorem Calculus: We state and prove Rolle's Theorem - if f(x) is continuous on [a, b], f is differentiable on (a, b), and f(a) = f(b), then there is an x in (a, b) with f'(x) = 0. The examples of (a) f(x) = x^2 -5x + 6 on [2, 3], and (b) f(x) = sin(x) on [pi/4, 3pi/4] are given.
Lecture 53 Play Video	Mean Value Theorem Calculus: We state and prove the Mean Value Theorem: If f is continuous on [a,b] and differentiable on (a,b), then there exists an x in (a, b) such that f'(x) = (f(b) - f(a))/(b-a). Example given are (a) f(x) = x^2 - 4x + 5 on [0, 3], and (b) f(x) = \|x\| on [-1. 3].
Lecture 54 Play Video	Increasing/Decreasing and Derivatives 1 Calculus: We define increasing and decreasing for a function on a region. Then we show that if the first derivative of f(x) is negative on a region, then f(x) is decreasing on that region. Examples are (a) f(x) = sqrt(x) on the nonnegative real numbers, and (b) f(x) = 1/x on the positive real numbers.
Lecture 55 Play Video	Increasing/Decreasing and Derivatives 2 Calculus: We give a procedure for finding the regions of increasing and decreasing for a function f(x) using the first derivative. Examples are f(x) = x^3 - 6x^2 and (b) f(x) = 1/x^2. Then the First Derivative Test for relative maxima and minima is given.
Lecture 56 Play Video	Example of Increasing/Decreasing 1 Calculus: Find the regions of increasing and decreasing for the functions: (a) f(x) = x^3 - 15x^2 + 27x, and (b) f(x) = (x^2 - 3)/(x-2). Then determine whether the critical points are local minimums, local maximums, or neither.
Lecture 57 Play Video	Example of Increasing/Decreasing 2 Calculus: Find the regions of increasing and decreasing for the following functions: (a) f(x) = tan(x), and (b) f(x) = sec(x) on (-pi/2, pi/2). Then determine if the critical points are locals minimums, local maximums, or neither.
Lecture 58 Play Video	Example of Increasing/Decreasing 3 Calculus: Find all critical points and regions of increasing and decreasing for the functions: (a) f(x) = sqrt(x^2 - 4), and (b) f(x) = 3 - \|x+2\|. Identify any local maxima or minima.
Lecture 59 Play Video	First/Second Derivative Test for f(x) = x^4 - 12x^3 Calculus: Using the First and Second Derivative Tests, find all relative maxima and minima for the function f(x) = x^4 - 12x^3.
Lecture 60 Play Video	First/Second Derivative Test for f(x) = sin(x) Calculus: Using the First and Second Derivative Test, find all relative maxima and minima of f(x) = sin(x).
Lecture 61 $First/Second Derivative Test for f(x) = x^2 - 6x^{4/3}$ Play Video	First/Second Derivative Test for f(x) = x^2 - 6x^{4/3} Calculus: Using the First and Second Derivative Tests, find all relative maxima and minima of the function f(x) = x^2 - 6x^{4/3}.
Lecture 62 Play Video	Concavity and the Second Derivative Calculus: We define concavity and explore its relation to the first and second derivatives. Inflection points are defined. Examples given are f(x) = x^3 - 12x and f(x) = x^{1/3}.
Lecture 63 Play Video	Concavity for f(x) = sin(x) Calculus: Using the second derivative, find the inflection points and regions of concavity for f(x) = sin(x).
Lecture 64 Play Video	Concavity for f(x) = (x^2 - 36)/(x-2) Calculus: Using the second derivative, find the inflection points and regions of concavity for f(x) = (x^2 - 36)/(x - 2).
Lecture 65 Play Video	Concavity for f(x) = \|x^2 - 4x - 12\| Calculus: Using the second derivative, find all inflection points and regions of concavity for the function f(x) = \|x^2 - 4x - 12\|.
Lecture 66 Play Video	Example of Limit at Infinity 1 Calculus: Find the limits at infinity of the function f(x) = (x^{-5} - x^{-2})/(x^{-3}- x^{-4}).
Lecture 67 Play Video	Example of Limit at Infinity 2 Calculus: Find the limits at +/- infinity of f(x) = (x - x^{-4/3))/(x^{-2/3} - x^{-1/3}).
Lecture 68 Play Video	Example of Limit at Infinity 3 Calculus: Find the limits at infinity of the function f(x) = sqrt(4x^2 + 1)/x. ADDED: One minute to check the answers.
Lecture 69 Play Video	Checklist for Sketching Functions Calculus: We give a checklist for sketching the graph of a function f(x) using the first and second derivatives. As an example, we sketch f(x) = 1/(x-2)^2.
Lecture 70 Play Video	Graph of f(x) = x^4 - 8x^3 Calculus: Using the first and second derivatives, sketch the graph of f(x) = x^4 - 8x^3.
Lecture 71 Play Video	Graph of f(x) = (x-2)/(x-1) Calculus: Using the first and second derivatives, sketch the graph of f(x) = (x-2)/(x-1).
Lecture 72 Play Video	Graph of f(x) = sin(x) + cos(x) Calculus: Using the first and second derivative, sketch the graph of f(x) = sin(x) + cos(x).
Lecture 73 Play Video	Graph of f(x) = sin(x)/(1+cos(x)) Calculus: Using the first and second derivatives, sketch the graph of f(x) = sin(x)/(1+cos(x)).
Lecture 74 $Graph of f(x) = x^{4/3} - 8x^{2/3}$ Play Video	Graph of f(x) = x^{4/3} - 8x^{2/3} Calculus: Using the first and second derivative, sketch the graph of f(x) = x^{4/3} - 8x^{2/3}
Lecture 75 Play Video	Optimization 1 Calculus: We present a procedure for solving word problems on optimization using derivatives. Examples include the fence problem and the minimum distance from a point to a line problem.
Lecture 76 Play Video	Optimization 2 Calculus: Given a 6 by 10 cm cardboard rectangle, find the box with the maximum volume if square are cut from the corners.
Lecture 77 Play Video	Optimization 3 Calculus: Given a wire 10cm long, we cut it and form a circle and a square. Find the cut that maximizes the sum of the areas.
Lecture 78 Play Video	Optimization - Maximizing Profit Calculus: If we sell concert tickets at a price of $60, we can expect an attendance of 300 people. For each $6 decrease in price, another 600 tickets are sold. Expenses are incurred at $12 per ticket. Compute the ticket price that maximizes profit.
Lecture 79 Play Video	Newton's Method 1 Calculus: Newton's Method uses tangent lines to approximate the zeros of a function. We estimate sqrt(3), derive the method, and note some issues with its application.
Lecture 80 Play Video	Newton's Method 2 Calculus: We use Newton's Method to approximate the solution to cos(x) = sin(x) between x=0 and x = pi/2.
Lecture 81 Play Video	Differentials 1 Calculus: The tangent line of f(x)=x^3 at x=1 is used to approximate (1.1)^3. The differential is defined, and we show how to use it to approximate the error in the estimate.
Lecture 82 Play Video	Differentials 2 Calculus: Use the differential to approximate: (a) (8.5)^{1/3}, (b) (1.9)^4. In essence, this is just the tangent line method of approximation.