dy dx dx dx as a BC Calculus 1 The Chain Rule is notation for a which says that we have the

Transcription

1 BC Calculus 1 The Chain Rule dy is notation for a which says that we have the for an expression set equal to (the dependent variable), where the variable is x. This is read dee why, dee or the derivative of y with to x. At this point we think of dy dy as. For reasons that we will soon discuss, we can, will, think of as a. f(x) = (4x 3 2x) 2 find f ' (x) f(x) = (4x 3 2x) 20 find f ' (x) f(x) = sin(3x 2 + 1) find f ' (x) Page 137 7, 13, 25

2 BC Calculus 2 A new approach to a new method For two reasons I tend not to use the notation that we used yesterday, when finding the derivative via the chain rule. Reason 1: It is really not. Reason : When the problems get more complex, like. Therefore I do The Chain Rule mostly in my. It is all about the word something. 2 3 f(x) sin (3x x), this notation becomes f(x) = sin(3x 2 + 1) find f ' (x) 3 2 f(x) 3x 2x 1 find f ' (x) With this approach even our original problem, f(x) = (4x 3 2x) 2 may be just as easily solved with The Chain Rule as it is by expanding. Notation for chain rule using Newtonian notation: If h(x) = f(g(x)) then h ' (x) = Page , 47, 53, 83, 95, 96, 97

3 BC Calculus 3 Implicit Differentiation y = 3x 2 + 6x 9 is an equation written in form; a form where the equation is solved for one variable, in terms of the other. In this case, y in terms of x. The notation for derivatives of equations written in explicit form makes. y = 3x 2 + 6x 9 y ' = 6x + 6 or even f(x) = sec(x) f ' (x) = sec(x) tan(x). What happens when the equation is not in explicit form? Such as 3x 2 + y 2 = 3x y 3 + 4? What is the name of the form this equation is written in? What does it mean to get the derivative of the equation 3x 2 + y 2 = 3x y 3 + 4? Not. In implicitly defined equations we have to be told more what to find, i.e. the derivative of what (variable) with respect to what variable. example: 3x 2 + y 2 = 3x y find dy. Now what? d What does sin(x) mean? So what we need to do is to write our question as a d question.

4 BC Calculus 4 example: 4x 2-3y = 2 + 4xy 2 find dy Page 146 1, 7, 11, 15, 28 The derivative of f(x) is written as ; the second derivative as. The derivative of y is written as ; the second derivative as. What if we write the first derivative of y as dy? What is the second derivative? d If expression tells us to find the derivative, then d tells us to find the derivative of a, or in other words, the derivative. This may help make sense of the above notation. example: 3x 2 + y 2 = 3x y find 2 dy 2. orthogonal Page , 53, 57, 59

5 BC Calculus 5 Related Rates A point is moving on the graph of y = 3x 3-5x in such a way that its x-coordinate is changing at the constant rate of 4 units per second. At what rate is y-coordinate changing, when the x-coordinate of the point is 2? Steps to solve related rates questions: 1. Draw 2. Label 3. Write 4. Differentiate 5. Solve

6 BC Calculus 6 A rock is dropped in a pond and the circular ripples created by the rock expand at a rate such that the area within the ripple(s) increases at a constant rate of 4 in 2 /min. When the ripple has a radius of 6 inches, find the rate at which the radius is changing. Page 154 7a, 13 (at x = 1), 20 A rocket is traveling vertically such that its height can be found by the equation h = 100t 2, where t is time, in seconds, from lift off and h is in feet. A camera, on the horizontal ground, 1200 feet from the launch site is tracking the rocket as it flies. At what rate is the angle of elevation of the camera changing, at the moment the rocket has been flying for 10 seconds? Page , 27a,b,c (use 7 ft. only), 30

7 BC Calculus 7 Coal is falling off of a conveyor belt and forming a conical pile. The height of the pile is always twice the diameter of the pile. If the coal is falling at the rate of 200 ft 3 /min, find that rate at which the height of the pile is increasing when the height is 100 feet. A five foot tall woman walks away from a 15 foot tall light pole (the light is at the top), at the rate of 4 feet/sec. At the moment the woman is 20 feet from the pole find how fast her shadow is growing. Find how fast the tip of her shadow is moving. Page , 25, 37

8 BC Calculus 8 Mo Related Rates 1. An oil tanker has an accident and oil pours out at the rate of 150 gallons per minute. Suppose that the oil spreads onto the water in a circle with a thickness of 1/10 of an inch. Given that 1 ft 3 equals 7.5 gallons, determine the rate at which the radius of the spill is increasing when the radius reaches 500 feet..767 ft/min 2. A car traveling west toward an intersection is ½ mile from the intersection, and traveling at the rate of 50 mph. Another car is traveling south and heading for the same intersection and is at that same moment ¼ mile from the intersection and traveling at the rate of 40 mph. Find the rate at which the distance between these two cars is changing at that moment mph 3. An artery has a cross-section that is a circle with radius 1.5 cm. Fat deposits are accumulating uniformly so that the artery opening decreases but remains a circle. Find the rate at which the artery opening (cross sectional area) is decreasing with respect to the thickness of the fat when the fat is.5 cm thick. 2π cm 2 /cm 4. A ball is thrown vertically such that its height can be found using the equation h(t) = -16t t, where the units are feet and seconds. A player, 25 feet from the throwing site when the ball is thrown, is running 5 ft/sec toward the throwing site. 1) Find the rate at which the distance between the player and the ball is changing 3 seconds after the ball was thrown. 2) Will the player reach the throwing/landing site before the ball hits the ground? ft/sec no