Unit 2 Lesson 1 - In this video we review modelling with linear functions, slope through 2 points, the point-slope and slope-intercept forms of a line, parallel and perpendicular lines and solving systems of equations. These are all extremely important for Calculus and are review of prior learning.
Unit 2 Lesson 2 - In this video I explore what a function is, the vocabulary and notation of functions, domain, range, symmetry, even functions, odd functions, piecewise and function composition.
Unit 2 Lesson 3 - There are four fundamental formulas in this lesson on exponential functions. In this video I briefly examine each.
Unit 2 Lesson 4 - In this video I define what an inverse function is and demonstrate two ways to find one. We introduce the concept of one-to-one functions and explain what they have to do with inverse functions. Then we define the logarithm function and use its properties as the inverse of the exponential function to solve an equation.
Unit 2 Lesson 5 - In this video I review the unit circle, evaluating trig functions and inverse trig functions, transforming trig functions, and solving trig equations.
Unit 3 Lesson 1 - In this video I demonstrate how to find the average rate of change and the instantaneous rate of change of displacement over time.
Unit 3 Lesson 2 - This video introduces the idea of a limit as presented in the context of finding instantaneous speed. We use a graphing calculator to talk about right-hand, left-hand and double-sided limits.
Unit 3 Lesson 3 - In this recording I examine two problems. In one, we look at a rational function with a vertical asymptote. We attempt to find the limit as x approaches that asymptote from both sides. Second, we examine two other rational functions and discover the limit as x approaches negative and positive infinity.
Unit 3 Lesson 3 End Behavior - In this recording we use long division to rewrite two rational functions and determine their end behavior. We verify this behavior on our graphing calculator.
Unit 3 Lesson 4 - In this recording I explain intuitively what it means for a function to be continuous. Then I define it formally. I do the same for a closed interval. Finally, I explain the (IVT) Intermediate Value Theorem and its implications, particularly in regard to finding x-intercepts or zeros of a function.
Unit 3 Lesson 5 - Here I find the average speed between two given times and then analyze the situation as the slope of a secant line to the function.
Unit 3 Lesson 6 - In this video I find the slope of the tangent line by examining the limit of a secant line, as the distance between the two points goes toward zero. I use Desmos to demonstrate this process graphically.
Unit 4 Lesson 1 - In this video I find the general derivative of y = x^3 using the standard and alternate definitions of a derivative.
Unit 4 Lesson 2 - Here I demonstrate how to graph the derivative of a function given the graph of the function. I then use Desmos to demonstrate the relationship between a polynomial function and its derivative. Finally, I demonstrate how to graph the original function given the graph of its derivative.
Unit 4 Lesson 3 - In this video I explain what it means for a function to be differentiable, that the limit definition is two-sided and must match for both the left hand and right-hand limits, and that this means there can be no gaps or cusps at or near the input value.
Unit 4 Lesson 4 - This video explains briefly what it means to be continuous on an interval, closed and open. It then solves a problem of missing parameters in a piecewise function that needs to be both continuous and differentiable.
Unit 4 Lesson 5 - This lesson is about the intermediate value theorem for derivatives. I briefly review IVT and then explain the implications of the IVT for derivatives. I then highlight the double-sided differential as an estimate for a derivative and then demonstrate how to find the derivative and numerical derivative in Desmos.
Unit 4 Lesson 7 - In this lesson I demonstrate the power rule, product rule, and quotient rule for derivatives.
Unit 4 Lesson 8 - In this video I define rectilinear motion. I then examine an example, finding the average velocity on a domain and the instantaneous velocity at a point. Then I examine the behavior of the object by looking at its velocity (the first derivative of position) and its acceleration (the second derivative of position).
Unit 4 Lesson 9 - In this video I start with the derivative of sine and cosine and derive the derivatives of the other four trig functions. I then briefly explain what jerk is. Finally, I find the equation of a line normal to a trig function at a given point.
Unit 6 Lesson 1 - In this lesson I explain what the chain rule is and demonstrate its use on three derivatives.
Unit 6 Lesson 2 - In this lesson I use the chain rule to find the equation of a line tangent to a given function, and another equation of a line normal to the function at the same point.
Unit 6 Lesson 3 - In this lesson I demonstrate implicit differentiation by finding an equation of a line tangent to a circle.
Unit 6 Lesson 4 - In this lesson I demonstrate how to find the second derivative implicitly. Note the two layers of substitution.
Unit 6 Lesson 5 - Here I demonstrate how to find the derivative of an inverse function by rewriting the inverse function without the inverse and differentiating implicitly. I use this technique to find the derivative of arcsine, arccosecant, and arctangent.
Unit 6 Lesson 6 - In this video I use implicit differentiation to derive the formula for d/dx a^x. I then use the formula to find the derivative of an exponential function.
Unit 6 Lesson 7 - In this video I use implicit differentiation to derive the formula for d/dx ln x. I then use it to find the derivative of a logarithmic function.
Unit 6 Lesson 8 - In this video I demonstrate how to differentiate functions with variables in the base and exponent by (1) taking the natural log of both sides and (2) differentiating implicitly, and (3) substituting back in the original expression for y.
Unit 7 Lesson 1 - In this video I demonstrate how to find extrema for a function by finding it's critical values (1) where the derivative is zero, (2) where the derivative does not exist, (3) at the end points of a closed domain.
Unit 7 Lesson 2 - In this video I explain the Mean Value Theorem (MVT) and what it means. I then demonstrate it in the case of f(x) = x^3 on [1,2].
Unit 7 Lesson 3 - In this video I use the first and second derivatives to determine local minima and maxima and the inflection point of a 3rd order polynomial function.
Unit 7 Lesson 5 - In this video I model creating a closed-top box from a sheet of paper and determine what dimensions should be cut from the paper to maximize the volume of the box.
Unit 7 Lesson 6 - In this video I briefly explain the concept of linearization around a point for the purpose of estimating values. I then derive Newton's formula for Newton's method of approximating a zero of a function and use it in Microsoft Excel to find the zeros of a function.
Unit 7 Lesson 7 - In this video I demonstrate how to find the rate of change of the volume of a cone given a specific height and rate of change of height of the same cone. This type of problem is best solved by (1) drawing a diagram, (2) eliminating any unwanted variables, (3) differentiating on time (dV/dt and dh/dt), and substituting any values given.
Unit 2 Lesson 2 - In this video I explore what a function is, the vocabulary and notation of functions, domain, range, symmetry, even functions, odd functions, piecewise and function composition.
Unit 2 Lesson 3 - There are four fundamental formulas in this lesson on exponential functions. In this video I briefly examine each.
Unit 2 Lesson 4 - In this video I define what an inverse function is and demonstrate two ways to find one. We introduce the concept of one-to-one functions and explain what they have to do with inverse functions. Then we define the logarithm function and use its properties as the inverse of the exponential function to solve an equation.
Unit 2 Lesson 5 - In this video I review the unit circle, evaluating trig functions and inverse trig functions, transforming trig functions, and solving trig equations.
Unit 3 Lesson 1 - In this video I demonstrate how to find the average rate of change and the instantaneous rate of change of displacement over time.
Unit 3 Lesson 2 - This video introduces the idea of a limit as presented in the context of finding instantaneous speed. We use a graphing calculator to talk about right-hand, left-hand and double-sided limits.
Unit 3 Lesson 3 - In this recording I examine two problems. In one, we look at a rational function with a vertical asymptote. We attempt to find the limit as x approaches that asymptote from both sides. Second, we examine two other rational functions and discover the limit as x approaches negative and positive infinity.
Unit 3 Lesson 3 End Behavior - In this recording we use long division to rewrite two rational functions and determine their end behavior. We verify this behavior on our graphing calculator.
Unit 3 Lesson 4 - In this recording I explain intuitively what it means for a function to be continuous. Then I define it formally. I do the same for a closed interval. Finally, I explain the (IVT) Intermediate Value Theorem and its implications, particularly in regard to finding x-intercepts or zeros of a function.
Unit 3 Lesson 5 - Here I find the average speed between two given times and then analyze the situation as the slope of a secant line to the function.
Unit 3 Lesson 6 - In this video I find the slope of the tangent line by examining the limit of a secant line, as the distance between the two points goes toward zero. I use Desmos to demonstrate this process graphically.
Unit 4 Lesson 1 - In this video I find the general derivative of y = x^3 using the standard and alternate definitions of a derivative.
Unit 4 Lesson 2 - Here I demonstrate how to graph the derivative of a function given the graph of the function. I then use Desmos to demonstrate the relationship between a polynomial function and its derivative. Finally, I demonstrate how to graph the original function given the graph of its derivative.
Unit 4 Lesson 3 - In this video I explain what it means for a function to be differentiable, that the limit definition is two-sided and must match for both the left hand and right-hand limits, and that this means there can be no gaps or cusps at or near the input value.
Unit 4 Lesson 4 - This video explains briefly what it means to be continuous on an interval, closed and open. It then solves a problem of missing parameters in a piecewise function that needs to be both continuous and differentiable.
Unit 4 Lesson 5 - This lesson is about the intermediate value theorem for derivatives. I briefly review IVT and then explain the implications of the IVT for derivatives. I then highlight the double-sided differential as an estimate for a derivative and then demonstrate how to find the derivative and numerical derivative in Desmos.
Unit 4 Lesson 7 - In this lesson I demonstrate the power rule, product rule, and quotient rule for derivatives.
Unit 4 Lesson 8 - In this video I define rectilinear motion. I then examine an example, finding the average velocity on a domain and the instantaneous velocity at a point. Then I examine the behavior of the object by looking at its velocity (the first derivative of position) and its acceleration (the second derivative of position).
Unit 4 Lesson 9 - In this video I start with the derivative of sine and cosine and derive the derivatives of the other four trig functions. I then briefly explain what jerk is. Finally, I find the equation of a line normal to a trig function at a given point.
Unit 6 Lesson 1 - In this lesson I explain what the chain rule is and demonstrate its use on three derivatives.
Unit 6 Lesson 2 - In this lesson I use the chain rule to find the equation of a line tangent to a given function, and another equation of a line normal to the function at the same point.
Unit 6 Lesson 3 - In this lesson I demonstrate implicit differentiation by finding an equation of a line tangent to a circle.
Unit 6 Lesson 4 - In this lesson I demonstrate how to find the second derivative implicitly. Note the two layers of substitution.
Unit 6 Lesson 5 - Here I demonstrate how to find the derivative of an inverse function by rewriting the inverse function without the inverse and differentiating implicitly. I use this technique to find the derivative of arcsine, arccosecant, and arctangent.
Unit 6 Lesson 6 - In this video I use implicit differentiation to derive the formula for d/dx a^x. I then use the formula to find the derivative of an exponential function.
Unit 6 Lesson 7 - In this video I use implicit differentiation to derive the formula for d/dx ln x. I then use it to find the derivative of a logarithmic function.
Unit 6 Lesson 8 - In this video I demonstrate how to differentiate functions with variables in the base and exponent by (1) taking the natural log of both sides and (2) differentiating implicitly, and (3) substituting back in the original expression for y.
Unit 7 Lesson 1 - In this video I demonstrate how to find extrema for a function by finding it's critical values (1) where the derivative is zero, (2) where the derivative does not exist, (3) at the end points of a closed domain.
Unit 7 Lesson 2 - In this video I explain the Mean Value Theorem (MVT) and what it means. I then demonstrate it in the case of f(x) = x^3 on [1,2].
Unit 7 Lesson 3 - In this video I use the first and second derivatives to determine local minima and maxima and the inflection point of a 3rd order polynomial function.
Unit 7 Lesson 5 - In this video I model creating a closed-top box from a sheet of paper and determine what dimensions should be cut from the paper to maximize the volume of the box.
Unit 7 Lesson 6 - In this video I briefly explain the concept of linearization around a point for the purpose of estimating values. I then derive Newton's formula for Newton's method of approximating a zero of a function and use it in Microsoft Excel to find the zeros of a function.
Unit 7 Lesson 7 - In this video I demonstrate how to find the rate of change of the volume of a cone given a specific height and rate of change of height of the same cone. This type of problem is best solved by (1) drawing a diagram, (2) eliminating any unwanted variables, (3) differentiating on time (dV/dt and dh/dt), and substituting any values given.