AP Calculus AB Chapter 4 Packet Implicit Differentiation. 4.5: Implicit Functions

Transcription

1 4.5: Implicit Functions We can employ implicit differentiation when an equation that defines a function is so complicated that we cannot use an explicit rule to find the derivative. EXAMPLE 1: Find dy dx for the following implicit functions. a) b) c) EXAMPLE 2: Suppose we have the differentiable function tangent line to the graph at.. Find the slope of the EXAMPLE 3: Find the slope of the tangent line to the curve at.

2 EXAMPLE 4: Find all points where the tangent lines to the graph of horizontal and vertical. are PRACTICE: In Exercises 1-4, find In Exercises 5-8, find an equation of the tangent line at the given point

3 9. A curve is given by the equation. a. Find. b. Write an equation for the line tangent to the curve at the point (1, 0). 10. Use implicit differentiation twice to find if. EXPANSION: 11. Sketch the circles y x and y ( x 3) 4. There is a line with positive slope that is 2 2 tangent to both circles. Determine the points at which this tangent line touches each circle. Show your work. 12. At pressure P (atmospheres), a certain fraction f of a gas decomposes. The quantities P and f are related by: a) Find df dp 4 1 f 2 f P 2 K where K is a constant.. b) Show that df dp 0 always. Explain this in practical terms.

4 Steps: 4.6: Related Rates of Change 1. Draw a picture. 2. Identify all given quantities and the quantities to be determined. Remember, the rate of change is the derivative with respect to time. 3. Write an equation relating the variables. You will probably need to use known formulas. 4. Differentiate both sides of your equation with respect to time (t). You will use the chain rule and implicit differentiation. Make sure you do not plug in any values for variables that are changing before you take the derivative. 5. After you have taken the derivative substitute all known values for the variables and their rates of change. 6. Solve the equation for the desired quantity. Example 1: A clown is inflating a balloon at a rate of increasing when ft? cubic ft/sec. At what rate is the radius Example 2: A painter is using a ladder 25 feet long which is leaning against a wall. His sadistic friend pulls the ladder away from the bottom of the wall at a rate of 14 ft/sec. At what rate is the top of the ladder moving down the wall when it is 7 feet above the ground?

5 Example 3: At a given moment, Mrs. Sapp is 30 miles north of an intersection, traveling toward it at 45 mph. At the same time Mr. Wichman is 40 miles east of the intersection, traveling away from it at 35 mph. Is the distance between Mrs. Sapp and Mr. Wichman increasing or decreasing at that moment? At what rate? Sapp Wichman Example 4: Water is flowing into a cone-shaped tank at the rate of 5 cubic in/sec. If the cone has altitude of 4 inches and a base radius of 3 inches, how fast is the water level rising when the water is 2 inches deep?

6 PRACTICE: 1. If find 2. Two automobiles start from a point A at the same time. One travels West at 80 miles per hour; the other travels North at 45 miles per hour. How fast is the distance between them increasing 3 hours after they start? 3. A spherical balloon is being inflated at the rate of 12. How fast is the radius changing at the moment when in?

7 4. A particle is moving along the curve in such a way that when,. Determine at that moment. 5. A ladder 15 feet tall leans against a vertical wall of a house. If the bottom of the ladder is pulled away horizontally from the house at 4 ft/sec, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 9 feet from the wall? 6. A cone (point down) with a height of 10 inches and a radius of 2 inches is being filled with water at the constant rate of 2. Determine how fast the water surface is rising when the water depth is 6 inches.

8 7. A streetlight is 15 feet above the sidewalk. A man 6 feet tall walks away from the light at a rate of 5 ft/sec. a) Determine a function relating the length of the man s shadow to his distance from the base of the streetlight. b) Determine the rate at which the man s shadow is lengthening at the moment he is 20 feet from the base of the light.

9 8. A ship is anchored 2 miles off a straight shore, and its searchlight is following a car that is moving along the shore at 40 miles per hour. How fast is the light turning (in radians per hour) when the car is 4 miles from the ship. 9. The length L of a rectangle is decreasing at the rate of while the width W is increasing at the rate of 2 cm/sec. When and, find the rate of change of a) the area b) the perimeter c) the length of a diagonal.

10 10. The function whose graph is sketched below gives the volume of air that a man has blown into a balloon after t seconds. Approximately how rapidly is the radius changing after 6 seconds? 4.7 Approximations In mathematics, when do we have to use approximations? Local Linearization (linear approx.): Example 1: What is the tangent line approximation of near? Example 2: Use the tangent line approximation of at to find an approximate value for.

11 Newton s Method 1 : Newton introduced a method for approximating zeros of differentiable functions. He introduced his method to find a zero of the function between and. In general, if we have calculated zero, the next approximation approximations of the actual, is given by: Example 3: Complete four iterations of Newton s Method for the function starting with an initial guess of =. 1 Newton s Method is not tested on the AP Calculus AB Exam.

12 PRACTICE: For # 1 and 2, find the local linearization of the given function at the given input value. 1. at 2. at 3. Estimate the value of the following expressions using a linear approximation. Then compute the difference between your estimate and the value given by your calculator. a) b)

13 Multiple Choice: AP PRACTICE 1. Given that, then dy dx a) b) c) d) e) 3x y y 3xy x x y y 3xy x x y y xy x x 3xy (3x 3 xy ) 2. A 26 foot ladder leans against a building so that its bottom moves away from the building at the rate of 3 ft/sec. When the bottom of the ladder is 10 ft from the building, the top is moving down at the rate of r ft/sec, where r is a) ft/sec b) 3/4 ft/sec c) 5/4 ft/sec d) 5/2 ft/sec e) 4/5 ft/sec 3. In the triangle below, the hypotenuse has fixed length 5 units, and is increasing at a constant rate of radians per min. At what rate is the area of the triangle increasing, in per minute, when h is 3 units? a) 1 b) 2 c) 3 d) 4 e) 5 θ 5 h