Implicit Differentiation Much of our algebraic study of mathematics has dealt with functions. In pre-calculus, we talked about two different types of equations that relate x and y explicit and implicit. Explicit equations in x and y are such that y is isolated on one side of the equation, and it equals an expression that is totally in terms of x. When I look at an explicitly defined equation, I can EXPLICITLY tell if it represents a function or not. In other words, when I see y 5 x, I know that for every x, there is only one y, making this equation a function. When I see y this equation not a function. 5 x, I know that for every x there are two y values, making Implicit equations are very different. Typically, they do not have y isolated on one side of the equation. Often, there is a power on the y term(s) in the equation and both y s and x s may appear throughout the equation. For example, the equation, x + y x + 4y +16 = 0, is that of a circle, if you will remember. Calculus can even be applied to implicitly defined equations. In this lesson, we will see how to differentiate those equations that are implicitly defined. dy We will utilize an alternate notation for the derivative. Instead of f '( x), we will use. As calculus dx was developed by two different men, a blend of their notations has been accepted. Let s think about how we differentiate y 5 x. Then, let s differentiate the implicit form of this equation, x + y = 5. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 304 Mark Sparks 01
Consider the graph of the circle to the right. Find the equation of the circle in implicit form below. Now, implicitly differentiate the equation of the circle in the space below dy Complete the table below finding the value of at each of the indicated points. Then, draw the dx graphical representation, the tangent line, on the graph at each indicated point. (0, ) (3, 3) (8, ) (3, 7) (6, ) (, ) (6, 6) Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 305 Mark Sparks 01
dy Find for each of the following implicitly defined equations. dx y x 3y y x e x y x 3 xy 3y x 5 3 y 3 x y Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 306 Mark Sparks 01
3 3 For what values of x will the curve x y 4xy 1have a horizontal tangent? Show your work and explain your thinking. 3 3 In terms of y, describe the values of x for which the curve x y 4xy 1will have a vertical tangent? Show your work and explain your thinking. Given the curve y y x 1, find d y dx. Given the curve x y 1, find d y dx. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 307 Mark Sparks 01
Related Rates Implicitly differentiate the following formulas with respect to time. State what each rate in the differential equation represents 1. A 4 r Surface Area of a Sphere. 3 V 4 3 r Volume of a Sphere 3. a c b, where c is is constant V 4. r h, where r is constant Volume of a Cylinder 5. x cos 15 V 6. 1 r h 3 Volume of a Cone Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 308 Mark Sparks 01
Air is leaking out of an inflated balloon in the shape of a sphere at a rate of 30π cubic centimeters per minute. At the instant when the radius is 4 centimeters, what is the rate of change of the radius of the balloon? 1. Identify all of the variables involved in the problem.. Identify which, if any, of the variables in the problem that remain constant. 3. Identify the rate(s) that are given and the rate that you wish to find. 4. Write an equation, often a geometric formula or trigonometric equation, that relates all of the variables in the problem for which a rate is given or for which a rate is to be determined. Substitute any value that represents a variable that is constant throughout the problem. It is important to keep in mind that you can have only one more variable than you have rates. You may have to make a substitution that relates one variable in terms of another. 5. Implicitly differentiate both sides of the equation with respect to time. 6. Substitute all instantaneous rates and values of the variable and solve for the remaining rate or variable. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 309 Mark Sparks 01
A stone is dropped into a calm body of water, causing ripples in the form of concentric circles. The radius of the outer ripple is increasing at a rate of 1 foot per second. When the radius is 4 feet, at what rate is the total area of the disturbed water changing? 1. Identify all of the variables involved in the problem.. Identify which, if any, of the variables in the problem that remain constant. 3. Identify the rate(s) that are given and the rate that you wish to find. 4. Write an equation, often a geometric formula or trigonometric equation, that relates all of the variables in the problem for which a rate is given or for which a rate is to be determined.. Substitute any value that represents a variable that is constant throughout the problem. It is important to keep in mind that you can have only one more variable than you have rates. You may have to make a substitution that relates one variable in terms of another. 5. Implicitly differentiate both sides of the equation with respect to time. 6. Substitute all instantaneous rates and values of the variable and solve for the remaining rate or variable. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 310 Mark Sparks 01
Water is leaking out of a cylindrical tank at a rate of 3 cubic feet per second. If the radius of the tank is 4 feet, at what rate is the depth of the water changing at any instant during the leak? 1. Identify all of the variables involved in the problem.. Identify which, if any, of the variables in the problem that remain constant. 3. Identify the rate(s) that are given and the rate that you wish to find. 4. Write an equation, often a geometric formula or trigonometric equation, that relates all of the variables in the problem for which a rate is given or for which a rate is to be determined. Substitute any value that represents a variable that is constant throughout the problem. It is important to keep in mind that you can have only one more variable than you have rates. You may have to make a substitution that relates one variable in terms of another. 5. Implicitly differentiate both sides of the equation with respect to time. 6. Substitute all instantaneous rates and values of the variable and solve for the remaining rate or variable. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 311 Mark Sparks 01
A cone has a diameter of 10 inches and a height of 15 inches. Water is being poured into the cone so that the height of the water level is changing at a rate of 1. inches per second. At the instant when the radius of the expose surface area of the water is inches, at what rate is the volume of the water changing? 1. Identify all of the variables involved in the problem.. Identify which, if any, of the variables in the problem that remain constant. 3. Identify the rate(s) that are given and the rate that you wish to find. 4. Write an equation, often a geometric formula or trigonometric equation, that relates all of the variables in the problem for which a rate is given or for which a rate is to be determined. Substitute any value that represents a variable that is constant throughout the problem. It is important to keep in mind that you can have only one more variable than you have rates. You may have to make a substitution that relates one variable in terms of another. 5. Implicitly differentiate both sides of the equation with respect to time. 6. Substitute all instantaneous rates and values of the variable and solve for the remaining rate or variable. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 31 Mark Sparks 01
A ladder 5 feet long is leaning against the wall of a house. The base of the ladder is pulled away from the wall at a rate of feet per second. a. How fast is the top of the ladder moving down the wall when the base of the ladder is 7 feet from the wall? b. Consider the triangle formed by the side of the house, the ladder, and the ground. Find the rate at which the area of the triangle is changing when the base of the ladder is 7 feet from the wall. c. Find the rate at which the angle formed by the ladder and the wall of the house is changing when the base of the ladder is 9 feet from the wall. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 313 Mark Sparks 01
An airplane is flying at an altitude of 5 miles and passes directly over a radar antenna. When the plane is 10 miles past the antenna, the rate at which the distance between the antenna and the plane is changing is 40 miles per hour. What is the speed of the plane? The radius of a sphere is increasing at a rate of inches per minute. Find the rate of change of the surface area of the sphere when the radius is 6 inches. A spherical balloon is expanding at a rate of 60π cubic inches per second. How fast is the surface area of the balloon expanding when the radius of the balloon is 4 inches. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 314 Mark Sparks 01