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Subject:	Mathematics
Topic:	Calculus
Views:	157,389

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Name:	integralCALC Academy (integralCALC)
Type:	Individual

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Instructor

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Krista King

Calculus I

Video Lectures

Displaying all 112 video lectures.

Lecture 1 Play Video	Vertical Line Test Overview In this video, Krista King from integralCALC Academy gives an overview for the Vertical Line Test.
Lecture 2 Play Video	Domain and Range In this video, Krista King from integralCALC Academy explains Domain and Range (Calculus problem example).
Lecture 3 Play Video	Finding Values and Domain and Range from a Graph In this video, Krista King from integralCALC Academy shows how to find points on the function and the domain and range of a function using only the graph of a function.
Lecture 4 Play Video	Equation Modeling In this video, Krista King from integralCALC Academy shows how to build an equation from a story problem. In this problem we use the description of an open top box to construct an equation for the cost of materials needed to build the box in terms of the width of the base of the box.
Lecture 5 Play Video	Modeling the Equation of a Piecewise Defined Function from its Graph In this video, Krista King from integralCALC Academy shows how to write the equation of a piecewise defined function, just by looking at the graph of the function. You'll write the equation of each piece individually, define the domain of each one, and then put them together to create the function.
Lecture 6 Play Video	Sketching a Graph from a Story Problem In this video, Krista King from integralCALC Academy shows how to sketch the graph of a function from a story problem. This word problem describes how temperature in a pot on a stove changes over time, and we sketch the graph of the temperature as a function of time.
Lecture 7 Play Video	Determining if a Function is Even, Odd or Neither In this video, Krista King from integralCALC Academy shows how to determine whether a function is an even function, an odd function, or neither an even nor odd function. We'll substitute -x for x, simplify, and then see whether our result is equal to our original function, and even, equal to our original function times -1, and odd, or neither.
Lecture 8 Play Video	How to Classify Functions In this video, Krista King from integralCALC Academy shows how to classify functions. We want to make sure to be as specific as possible, but still ensure that we classify the entire function, and not just a piece of it.
Lecture 9 Play Video	Equation of the Line in Slope-intercept Form In this video, Krista King from integralCALC Academy shows how to use the formula for the equation of a line, and various pieces of information about the line, to find the equation of the line in slope-intercept form. You can easily find the equation of the line, as long as you have two pieces of information about the line, which could be any combination of a point, another point, a slope, an intercept, or another line that is parallel or perpendicular to the line you're trying to solve for.
Lecture 10 Play Video	Equation of a Line in Point-slope Form In this video, Krista King from integralCALC Academy shows how to use the formula for the equation of the line in point-slope form to find the equation of the line given various pieces of information about the line, including a slope and a y-intercept, a slope and a point, two points, or a point and the fact that the line is parallel to another line.
Lecture 11 Play Video	Least Squares Line In this video, Krista King from integralCALC Academy shows how to find the equation of the least squares line, also known as the line of least square, the regression line, or the line of regression. Given several coordinate points, use formulas for m, the slope, and b, the y-intercept, to calculate the equation of the line that is the average of all the points.
Lecture 12 Play Video	Finding the Inverse of a Function In this video, Krista King from integralCALC Academy shows how to finding the inverse of a function (Calculus problem example).
Lecture 13 Play Video	Sketch the Graph of a Parabola In this video, Krista King from integralCALC Academy shows how to sketch the graph of a parabola. You'll identify the form of the equation as standard form or an alternative form, then find the vertex and two other points on the parabola, then sketch its graph. The two additional points should be the y-intercept and its associated point on the other side of the axis of symmetry, which runs through the vertex of the parabola.
Lecture 14 Play Video	Finding the Center and Radius of the Circle In this video, Krista King from integralCALC Academy shows how to find center and radius of the circle (Calculus problem example).
Lecture 15 Play Video	Sketch the Graph of a Circle In this video, Krista King from integralCALC Academy shows how to sketch the graph of a circle. You'll complete the square with respect to x and y to change the equation of the circle into standard form. Then you'll identify the center of the circle at the point (h,k), and find the radius of the circle, r. Using the center and radius of the circle, sketch its graph and find the left-most, right-most, top and bottom of the circle.
Lecture 16 Play Video	Combinations of Functions and their Domains In this video, Krista King from integralCALC Academy show how to calculate combinations of two functions, then find the domain of the combination function.
Lecture 17 Play Video	Composite Functions In this video, Krista King from integralCALC Academy show how to calculate the composite of three functions, f(x), g(x) and h(x) by plugging h(x) into g(x) and then your result into f(x).
Lecture 18 Play Video	Composite Functions and their Domains In this video, Krista King from integralCALC Academy shows how to calculate the compositions of two functions, including f(g(x)), g(f(x)), f(f(x)), and g(g(x)). Then describe the domain of each composite function.
Lecture 19 Play Video	Describing Transformations Algebraically In this video, Krista King from integralCALC Academy shows how to describe transformations of functions algebraically, including shifting up, shifting down, shifting left and right, reflecting about the x-axis, reflecting about the y-axis, and vertical stretching and shrinking.
Lecture 20 Play Video	Graphing Transformations In this video, Krista King from integralCALC Academy shows how to graph transformations of a function. We'll look at vertical shifts, reflections about the x and y axis, and vertical stretching and shrinking.
Lecture 21 Play Video	Using Transformations to Sketch a Graph In this video, Krista King from integralCALC Academy shows how to use transformations to sketch a graph, one piece at a time.
Lecture 22 Play Video	Determine Whether a Function is 1 to 1 In this video, Krista King from integralCALC Academy shows how to determine whether or not a function is 1-to-1.
Lecture 23 Play Video	Find the Inverse of a Function and Sketch its Graph In this video, Krista King from integralCALC Academy shows how to use algebra to find the inverse of a function, then sketch the graph of the inverse function by reflecting the original graph over the line y=x.
Lecture 24 Play Video	Finding the Linear Function Given Two Points on its Inverse In this video, Krista King from integralCALC Academy shows how to find the equation of a line, given two points on the linear function's inverse.
Lecture 25 Play Video	Use Laws of Logarithms to Simplify a Logarithmic Function In this video, Krista King from integralCALC Academy shows how to use laws of logarithms to simplify a logarithmic function.
Lecture 26 Play Video	Use the Quadratic Formula to Find Roots of the Function In this video, Krista King from integralCALC Academy shows how to use the quadratic formula to find the roots of a quadratic function. To do this, make sure that you can't first factor the function to find the roots. Then identify coefficients a, b and c from the standard form of the quadratic function, and plug them into your quadratic formula.
Lecture 27 Play Video	Completing the Square of a Quadratic Function In this video, Krista King from integralCALC Academy shows how to use a formula to complete the square of a quadratic function. You'll need to identify whether the quadratic function is set equal to a constant, then choose which formula to use. Set the quadratic function equal to the formula, then expand and collect terms on the right-hand side. Finally, equate coefficients on both sides of the equation to solve for variables in the formula.
Lecture 28 Play Video	Polynomial Long Division for Rational Functions In this video, Krista King from integralCALC Academy shows how to use polynomial long division to simplify a rational function, which is the quotient, or fraction, of two polynomials. This will work the same way as long division with real numbers. The numerator goes on the inside of the long division problem; the denominator goes on the outside. Figure out what you have to multiply by the first term from the numerator to get the first term from the denominator. Put this above the first term from the denominator, then multiply this value by the entire denominator, writing the result below the numerator. Subtract the value from the numerator, carry down the next value, and repeat this process.
Lecture 29 Play Video	Hyperbolic Identities In this video, Krista King from integralCALC Academy shows how to use hyperbolic identities to simplify a hyperbolic function.
Lecture 30 Play Video	Limits: Substitution Method In this video, Krista King from integralCALC Academy gives an example of the Substitution Method for limits.
Lecture 31 Play Video	Limits: Factoring Method In this video, Krista King from integralCALC Academy gives an example of the Factoring Method for limits.
Lecture 32 Play Video	Limits: Conjugate Method In this video, Krista King from integralCALC Academy gives an example of the Conjugate Method for limits.
Lecture 33 Play Video	Use Limit Laws to Evaluate Limits of Combination Functions In this video, Krista King from integralCALC Academy shows how to use limit laws to evaluate limit problems, specifically combinations of functions.
Lecture 34 Play Video	Limits: Crazy Graphs In this video, Krista King from integralCALC Academy goes through examples of crazy graphs involving limits.
Lecture 35 Play Video	Limits at Infinity In this video, Krista King from integralCALC Academy shows how to find the limit of a function as x approaches infinity. In other words, if we take x to be infinitely large, what value does the function get close to? To figure this out, you'll need to use algebra to simplify the function, then use L'Hospital's rule several times until the function is simplified enough that you can evaluate it at infinity.
Lecture 36 Play Video	Infinite Limits In this video, Krista King from integralCALC Academy shows how to calculate an infinite limit. In order to determine whether or not the function has an infinite limit as x approaches a specific point, you'll need to first confirm that your function as a vertical asymptote and the point you're approaching. The function will have a vertical asymptote where it is undefined because the denominator of the function is equal to zero. Then you'll need to plug in values close to and on both sides of the number you're approaching. If the value of the function gets very large as you approach the vertical asymptote, then the limit is positive infinity. If the values get very large but negative as you approach the vertical asymptote, then the limit is negative infinity. Remember that infinite limits don't technically exist, but because it gives us so much more information about the function to say that the limit is positive of negative infinity, we'll often provide that answer instead of stating that the limit does not exist (DNE). If the one-sided limits are not equal, there is no general limit.
Lecture 37 Play Video	Limits: Trigonometric (Example 2) In this video, Krista King from integralCALC Academy goes through a trigonometric limits problem.
Lecture 38 Play Video	Limits: One-Sided In this video, Krista King from integralCALC Academy explains One-Sided limits (Calculus problem example).
Lecture 39 Play Video	How to Prove that the Limit Does Not Exist In this video, Krista King from integralCALC Academy shows how to prove that the limit does not exist. Investigate the existence of the left-hand limit, the right-hand limit, and then whether or not the left-hand limit is equal to the right-hand limit.
Lecture 40 Play Video	Precise Definition of the Limit In this video, Krista King from integralCALC Academy talks about the Precise Definition of the Limit.
Lecture 41 Play Video	Finding Delta from a Graph and the Epsilon-delta Definition of the Limit In this video, Krista King from integralCALC Academy shows how to use the graph of a function, and the epsilon-delta definition of the limit (precise definition of the limit), to find delta, given epsilon as a constraint.
Lecture 42 Play Video	Squeeze Theorem In this video, Krista King from integralCALC Academy explains the squeeze theorem (Calculus problem).
Lecture 43 Play Video	Limit of an Inequality with Squeeze Theorem In this video, Krista King from integralCALC Academy shows how to find the limit of a function that's sandwiched between two other functions, in an inequality. See how this results in use of the squeeze theorem.
Lecture 44 Play Video	Continuity In this video, Krista King from integralCALC Academy explains continuity (Calculus problem example).
Lecture 45 Play Video	Removable Discontinuity In this video, Krista King from integralCALC Academy explains removable discontinuity (Calculus problem example).
Lecture 46 Play Video	Finding the Value that Makes the Function Continuous In this video, Krista King from integralCALC Academy explains how to find the value that makes the function continuous.
Lecture 47 Play Video	Intermediate Value Theorem Overview In this video, Krista King from integralCALC Academy gives an overview for the Intermediate Value Theorem.
Lecture 48 Play Video	Intermediate Value Theorem to Prove a Root in an Interval In this video, Krista King from integralCALC Academy shows how to use the intermediate value theorem to prove that a function has a root (point of intersection with the x-axis) on a given interval. To prove that the root exists in the interval, plug the left and right endpoints of the interval into the function. If the left endpoint produces a negative value and the right endpoint produces a positive value, then you can use the intermediate value theorem to prove that the graph crosses the x-axis (has a value of zero) somewhere between the endpoints.
Lecture 49 Play Video	Prove the Equation Has at Least One Real Root In this video, Krista King from integralCALC Academy shows how to use the Intermediate Value Theorem to prove that the function as at least one real root. To do this, you'll need to look at the range of each term in the function, and then use that information to pinpoint values you should focus on. You'll need to look for points above and below the x-axis. If you can prove that a point exist below the x-axis, and that another point exists above the x-axis, and you know that the function is continuous between those points, then you can use the Intermediate Value Theorem to prove that the function must assume a value of zero somewhere between those two points. In other words, the function must have a real root between these points.
Lecture 50 Play Video	How to Calculate the Difference Quotient In this video, Krista King from integralCALC Academy shows how to calculate the difference quotient of a function at a particular point. Use the difference quotient formula.
Lecture 51 Play Video	Power Rule In this video, Krista King from integralCALC Academy shows how to use power rule to calculate the derivative of a function. In order to complete this problem, you'll need to understand the power rule formula, and then use it to evaluate the derivative of the function. For this particular problem, you'll have to simplify some square roots before you use the power rule.
Lecture 52 Play Video	Derivatives of Linear Combinations In this video, Krista King from integralCALC Academy gives an overview for the derivative of a linear combination.
Lecture 53 Play Video	Product Rule In this video, Krista King from integralCALC Academy shows how to use product rule to calculate the derivative of a function. In order to complete this problem, you'll need to understand the product rule formula, and then use it to evaluate the derivative of the function. For this particular problem, you'll have to eliminate fractions by bringing power functions from the denominators into the numerators, and changing positive exponents into negative exponents.
Lecture 54 Play Video	Product Rule - 3+ Functions In this video, Krista King from integralCALC Academy shows how to use product rule to calculate the derivative of a function which is the product of three or more functions.
Lecture 55 Play Video	Quotient Rule In this video, Krista King from integralCALC Academy shows how to use quotient rule to calculate the derivative of a function. In order to complete this problem, you'll need to understand the quotient rule formula, and then use it to evaluate the derivative of the function. For this particular problem, you'll have to deal with a constant, as well as some fractions that you'll need to turn into power functions with negative exponents.
Lecture 56 Play Video	Reciprocal Rule In this video, Krista King from integralCALC Academy shows how to use reciprocal rule to calculate the derivative of a function. In order to understand the formula, we'll derive it from the quotient rule formula, then use the reciprocal rule formula to find the derivative of our function.
Lecture 57 Play Video	Chain Rule In this video, Krista King from integralCALC Academy talks about the Chain Rule (Calculus example).
Lecture 58 Play Video	Chain Rule for Derivatives with Product Rule In this video, Krista King from integralCALC Academy shows how to use the chain rule to calculate the derivative of the product of two functions. To use chain rule, you'll need to identify an inside and outside function. When you take the derivative of the function in general, you'll take the derivative of the outside function first, leaving the inside function completely untouched, then you'll multiply your result by the derivative of the inside function.
Lecture 59 Play Video	Chain Rule for Derivatives with Quotient Rule In this video, Krista King from integralCALC Academy shows how to use chain rule to find the derivative of a function involving cosine. To use chain rule, you'll identify inside and outside functions, then take the derivative of the outside function, leaving the inside function completely untouched. Then you'll multiply your result by the derivative of the inside function. In this particular problem, you'll need to use chain rule in conjunction with quotient rule to find the derivative. Once you've got all the pieces plugged in, simplify your answer as much as possible.
Lecture 60 Play Video	Chain Rule for Derivatives with Trig Functions In this video, Krista King from integralCALC Academy show how to use Chain Rule to take the derivative of a function. In order to do this, you'll need to identify inside and outside functions. When you take the derivative, you'll take the derivative of the outside function first, leaving the inside function completely untouched. Ignore it, as if it were simple. Think about it like peeling away layers of an onion, or just as working your way in from the outside. After you've taken the derivative of the outside function, multiply that result by the derivative of the inside function. Then simplify your result as much as possible.
Lecture 61 Play Video	Trigonometric Derivatives: Overview In this video, Krista King from integralCALC Academy gives an overview for trigonometric derivatives.
Lecture 62 Play Video	Trigonometric Derivatives (Example 1) In this video, Krista King from integralCALC Academy talks about trigonometric derivatives (Calculus problem example).
Lecture 63 Play Video	Derivatives of Inverse Trig Functions: arcsin In this video, Krista King from integralCALC Academy shows how to calculate the derivative of an inverse trig function. In this particular example, we'll calculate the derivative of arcsin, the inverse sine function.
Lecture 64 Play Video	Derivatives of Inverse Trig Functions: arccot In this video, Krista King from integralCALC Academy shows how to calculate the derivative of an inverse trig function. In this particular example, we'll calculate the derivative of arccot, the inverse cotangent function.
Lecture 65 Play Video	Derivatives of Hyperbolic Functions In this video, Krista King from integralCALC Academy talks about derivatives of hyperbolic functions (Calculus problem example).
Lecture 66 Play Video	Derivative of an Inverse Hyperbolic Function In this video, Krista King from integralCALC Academy shows how to prove an inverse hyperbolic identity.
Lecture 67 Play Video	Derivatives of Natural Logs (ln) In this video, Krista King from integralCALC Academy explains the derivatives of natural logs (Calculus example).
Lecture 68 Play Video	Use Laws of Logarithms to Find the Derivative In this video, Krista King from integralCALC Academy explains the laws of logarithms (Calculus problem example).
Lecture 69 Play Video	Derivatives of Exponentials (e^x) In this video, Krista King from integralCALC Academy explains derivatives of exponentials (Calculus example).
Lecture 70 Play Video	Equation of the Tangent Line In this video, Krista King from integralCALC Academy talks about the equation of the tangent line (Calculus example).
Lecture 71 Play Video	Differentiability and Vertical Tangent Lines In this video, Krista King from integralCALC Academy explains a differentiability and vertical tangent line (Calculus problem example).
Lecture 72 Play Video	Equation of the Normal Line at a Point In this video, Krista King from integralCALC Academy shows how to find the equation of the normal line at a given point. To find the equation of the normal line, you'll need to first calculate the derivative of the function, then plug the given point into the derivative to find the slope of the tangent line. Plug the slope and the given point into the point-slope formula for the equation of the line to find the equation of the tangent line. Then take the negative reciprocal of the slope of the tangent line to find the slope of the normal line, which is the line perpendicular to the tangent line. Finally, plug the new slope and the given point into the point-slope formula for the equation of the line to find the equation of the normal line.
Lecture 73 Play Video	Average Rate of Change In this video, Krista King from integralCALC Academy looks at our first example for Average Rate of Change.
Lecture 74 Play Video	Implicit Differentiation In this video, Krista King from integralCALC Academy explains implicit differentiation using an example.
Lecture 75 Play Video	Use Implicit Differentiation to Find the Equation of the Tangent Line at a Point In this video, Krista King from integralCALC Academy shows how to use implicit differentiation to calculate the equation of the tangent to the curve at a specific point. Use implicit differentiation to find the first derivative of y, or y', or y prime, then plug the given point into the first derivative to get the slope of the tangent line, m. Plug the slope and the given point into the point-slope formula for the equation of the line, and simplify to get the equation of the tangent line to the curve at that point.
Lecture 76 Play Video	Use Implicit Differentiation to Find the Second Derivative of y (y'') In this video, Krista King from integralCALC Academy shows how to use implicit differentiation to calculate the second-derivative of y, or y''. You'll calculate the first derivative of y, then solve for y', and then calculate the second derivative of y, or y''.
Lecture 77 Play Video	Half Life In this video, Krista King from integralCALC Academy talks about Half Life (Calculus problem example).
Lecture 78 Play Video	Continuously Compounded Interest In this video, Krista King from integralCALC Academy talks about continuously compound interest (Calculus problem example).
Lecture 79 Play Video	Sales Decline In this video, Krista King from integralCALC Academy talks about Sales Decline (Calculus problem example).
Lecture 80 Play Video	Linear Approximation in One Variable In this video, Krista King from integralCALC Academy explains a linear approximation in one variable (Calculus problem).
Lecture 81 Play Video	Linearization of a Function at a Point In this video, Krista King from integralCALC Academy shows how to find the linearization, or linear approximation of a function at a point. Find the value of the function at the given point, then find the value of the first derivative of the function at the given point, then plug both values and the given point into the linearization formula and simplify.
Lecture 82 Play Video	Critical Points In this video, Krista King from integralCALC Academy shows how to find the critical points of a function, and understand what they are. To find the critical points, you'll need to take the derivative of the given function, set that equal to 0, and then factor in order to solve for x.
Lecture 83 Play Video	Increasing and Decreasing (Example 1) In this video, Krista King from integralCALC Academy talks about increasing and decreasing (Calculus example).
Lecture 84 Play Video	Concavity and Inflection Points In this video, Krista King from integralCALC Academy talks about concavity and inflection points (Calculus problem example).
Lecture 85 Play Video	First Derivative Test In this video, Krista King from integralCALC Academy talks about the first derivative test.
Lecture 86 Play Video	Second Derivative Test: One Variable In this video, Krista King from integralCALC Academy talks about the second derivative test.
Lecture 87 Play Video	Vertical Asymptotes: Overview In this video, Krista King from integralCALC Academy gives an overview for vertical asymptotes.
Lecture 88 Play Video	Horizontal Asymptotes: Basic Overview In this video, Krista King from integralCALC Academy gives a basic overview of horizontal asymptotes.
Lecture 89 Play Video	Horizontal Asymptotes: Further Detail In this video, Krista King from integralCALC Academy goes into further detail about horizontal asymptote rules.
Lecture 90 Play Video	Slant Asymptotes In this video, Krista King from integralCALC Academy explains a slant asymptote.
Lecture 91 Play Video	Sketching Graphs (Example 1) In this video, Krista King from integralCALC Academy talks about sketching graphs.
Lecture 92 Play Video	Maxima and Minima on a Closed Range In this video, Krista King from integralCALC Academy talks about Maxima and Minima (Calculus example).
Lecture 93 Play Video	Dimensions that Minimize the Surface Area of a Cylinder In this video, Krista King from integralCALC Academy show how to find the dimensions of a cylinder that will minimize its surface area. Since this is an optimization equation, draw a picture of the problem and write down what you know, identify optimization and constraint equations, and then solve the constraint equation for one of the variables so that you can plug your answer into the optimization equation and get the optimization equation in terms of one variable. Then take the derivative of the optimization equation, set it equal to zero, and solve for the variable. Then make sure you're answering the question you were actually asked.
Lecture 94 Play Video	Largest Area of a Rectangle Inscribed in a Semicircle In this video, Krista King from integralCALC Academy shows how to find the largest area of a rectangle that can be inscribed inside a semicircle, given that the semicircle has radius r. Since this is an optimization problem, draw a picture of the problem and write what you know. Then identify optimization and constraint equations. Solve the constraint equation for one of the variables, and plug the solution into the optimization equation to get the optimization equation in terms of one variable. Then take the derivative of the optimization equation, set it equal to zero, and solve for the variable. Make sure to answer the question you were asked!
Lecture 95 Play Video	Dimensions that Maximize the Area of the Rectangle In this video, Krista King from integralCALC Academy shows how to calculate the derivative of a parametric curve and find the equation of the tangent line to the parametric curve at a given point t.
Lecture 96 Play Video	Largest Possible Volume of a Cylinder Inscribed in a Sphere In this video, Krista King from integralCALC Academy shows how to find the largest possible volume of a cylinder inscribed in a sphere with radius r: "Find the largest possible volume of a right circular cylinder that can be inscribed in a sphere with radius r." To solve this optimization problem, draw a picture of the problem and label all parts of the diagram, then write down everything you know. Next, identify optimization and constraint equations. The optimization equation will be the equation for the volume of the cylinder, since the goal is to maximize the volume of the cylinder. The constraint equation will include the variable that constrains you. In this case, the constraint equation will be the equation for the radius of the sphere. Solve the constraint equation for one of the variables, and then plug the result into the optimization equation. Then simplify the optimization equation, take its derivative, and set it equal to zero. Solve for the variable, and then plug that back into the equation for the volume of the cylinder to find the largest possible volume of the cylinder that can be inscribed in a sphere with radius r.
Lecture 97 Play Video	Maximum Volume of a Cone Shaped Cup In this video, Krista King from integralCALC Academy shows how find the largest possible volume of a cone-shaped up made from a circular piece of paper with radius R, where a sector has been removed and sides CA and CB are joined together. To complete this optimization problem, you'll need to draw a picture of the problem and write down what you know. Then you'll imagine that you'll take a vertical slice of the cone shaped cup so that you can use the pythagorean theorem to relate the radius of the paper to the radius and height of the cone. Solve the pythagorean theorem, the constraint equation, for one of the variables so that you can plug it into the optimization equation, which will be the equation for the volume of the cone. Then simplify the equation for the volume, take the derivative and set it equal to 0 to solve for the height. Plug the height back into the pythagorean theorem to find the radius, then plug both values back into the volume equation to find the volume.
Lecture 98 Play Video	Dimensions of the Rectangle with Largest Area Inscribed in an Equilateral Triangle In this video, Krista King from integralCALC Academy shows how to find the dimensions of the rectangle that maximize its area, assuming the rectangle is inscribed in an equilateral triangle of side L. In order to complete this optimization problem, you'll need to draw a picture of the problem and write down everything you know, identify optimization and constraint equations, and then use the derivative of the optimization equation to find the dimensions. The optimization equation will be an equation for the area of the rectangle in terms of its base and height. If you draw the equilateral triangle and the rectangle on an xy coordinate plane with their bases on the x-axis and the figures symmetric about the y-axis, you can describe the right-edge of the base as extending to the coordinate (x,0). Therefore, x is equal to half of the base, or b/2. Then, use the method of similar triangles and the Pythagorean theorem to relate the base and height of the smaller triangle with the larger triangle. This allows you to solve for h. Plug the value you got for the height and half the base back into your area equation. Then simplify it, take its derivative, set the derivative equal to 0, and solve for x. This will lead you to the length of the base. Then you can plug this into your height equation to find the height of the rectangle.
Lecture 99 Play Video	Applied Optimization: Two Real Numbers with Difference 20 and Minimum Possible Product In this video, Krista King from integralCALC Academy explains an applied optimization example where we find two real numbers with difference 20 and minimum possible product.
Lecture 100 Play Video	Applied Optimization: Area and Margins of a Page In this video, Krista King from integralCALC Academy talks about applied optimization, area and margins of a page.
Lecture 101 Play Video	Related Rates: Radius of a Balloon and Changing Price In this video, Krista King from integralCALC Academy explains related rates using the example of a balloon.
Lecture 102 Play Video	Related Rates: Water Level in a Tank In this video, Krista King from integralCALC Academy explains related rates using the example of a water tank.
Lecture 103 Play Video	Related Rates: Ladder Sliding Down a Wall In this video, Krista King from integralCALC Academy talks about Related Rates, using the example of a ladder.
Lecture 104 Play Video	Related Rates: Distance Between Observer and Airplane In this video, Krista King from integralCALC Academy talks about Related Rates, using the example of an airplane.
Lecture 105 Play Video	Mean Value Theorem In this video, Krista King from integralCALC Academy explains the Mean Value Theorem (Calculus problem example).
Lecture 106 Play Video	Rolle's Theorem In this video, Krista King from integralCALC Academy explains Rolle's Theorem (Calculus problem example).
Lecture 107 Play Video	Newton's Method In this video, Krista King from integralCALC Academy explains Newton's Method (Calculus problem example).
Lecture 108 Play Video	L'Hopital's Rule In this video, Krista King from integralCALC Academy explains L'Hopital's Rule (Calculus problem).
Lecture 109 Play Video	Position Function In this video, Krista King from integralCALC Academy explains the position function (Calculus problem example).
Lecture 110 Play Video	All About a Particle's Position Function In this video, Krista King from integralCALC Academy shows how to identify a particle's position function, its first derivative which is its velocity function, and its second derivative which is its acceleration function.Find velocity at time t, and find velocity after 2 seconds and after 4 seconds. Determine when the particle is at rest by setting the velocity function equal to 0 and solving for t. Find out when the particle is moving forward by identifying where the velocity function is positive. Draw a diagram of the particle's motion as it moves along the speed axis, then use the diagram to determine the distance traveled by the particle in during the first 5 seconds. Find the acceleration at time t by calculating the second derivative of the position function, and then figure out the acceleration after 4 seconds. Calculate when the particle is speeding up and when it's slowing down by finding out where the velocity function is positive and negative, where the acceleration function is positive and negative, and then comparing the points in time at which both velocity and acceleration are positive, and where both velocity and acceleration are negative. When they are both positive or both negative (have the same sign), the particle will be speeding up. Whenever they have opposite signs, it means that the particle is slowing down.
Lecture 111 Play Video	Vertical Motion (Differentiation) In this video, Krista King from integralCALC Academy explais vertical motion (Calculus problem example).
Lecture 112 Play Video	Marginal Cost, Revenue and Profit In this video, Krista King from integralCALC Academy talks about Marginal Cost, Revenue and Profit (Calculus problem example).