Chapter 2 THE DERIVATIVE

Chapter 2 THE DERIVATIVE

2.1 Two Problems with One Theme

Tangent Line (Euclid) A tangent is a line touching a curve at just one point. - Euclid (323 285 BC)

Tangent Line (Archimedes) A tangent to a curve at P is the line that best approximates the curve near P. - Archimedes (227 212 BC)

Examples 1. Find the slope of the tangent lines to the curve of y = f x = x 2 + x + 2 at the points with x-coordinates 1, 1 2, 2,3. 2. Find the equation of the tangent line to the curve y = 1 x at 2, 1 2.

Average Velocity and Instantaneous Velocity An object P falls in a vacuum. Experiment shows that if it starts from rest, P falls 16t 2 feet in t seconds. The average velocity of P during the interval [1,2] is 16 2 2 16 2 1 The average velocity of P during the interval [1,1.5] is 16 1.5 2 16 1.5 1 The average velocity of P during the interval [1,1.1] is 16 1.1 2 16 1.1 1 = 48. The average velocity of P during the interval [1,1.01] is 16 1.01 2 16 1.01 1 = 40. = 33.6. = 32.16.

Example An object, initially at rest, falls due to gravity. a. Find its instantaneous velocity at t = 3.8 seconds. b. How long will it take the object to reach an instantaneous velocity of 112 feet per second?

2.2 The Derivative

The Derivative If the limit exists, we say f is differentiable at x. Finding a derivative is called differentiation and the part of Calculus associated with derivatives is called Differential Calculus. Examples. 1. Let f x = x 3 + 7x. Find f 4. 2. Find F x if F x = x, x 0.

Equivalent Forms for Derivatives Examples. Each of the following is a derivative, but of what function and at which point? a. lim b. lim x 3 4+h 2 16 h 0 h 2 x 2 3 x 3

Differentiability Implies Continuity Proof? The converse of Theorem A is not true. A continuous function is not differentiable at any point where the graph of the function has sharp corner.

Increments and Leibniz Notation The change in x is called an increment of x and denoted by x. Corresponding to the increment of x, we have an increment of y, y.

The Graph of Derivative The derivative f (x) gives the slope of the tangent line of the graph y = f(x) at the value of x. Example. Given the following graph of y = f(x), sketch the graph of y = f (x)

2.3 Rules for Finding Derivatives

Derivative as an Operator Three notations for derivative: f (x) or D x f(x) or dy dx

The Constant and Power Rules

D x is a Linear Operator Example. Find the derivative of 4x 6 2x 4 + 6x 3 x 2 + x 101.

Product and Quotient Rules Is the derivative of a product, the product of the derivatives?

Examples 1. Find D x 3x 2 5 6x 4 + 2x 2. Find D x y if y = 2 x 4 +1 + 3 x 3. Show that D x x n = nx n 1

2.4 Derivatives of Trigonometric Function

The Derivatives of sin x and cos x Examples. 1. Find D x x 2 sin x. 2. Find the equation of the tangent line to the graph of y = 3 sin x at the point π, 0.

The Derivatives of Other Trigonometric Functions Examples. 1. Find D x x n tan x. 2. Find all points in the graph y = sin 2 x where the tangent line is horizontal.

2.5 The Chain Rule

Derivative for a composite function Derivative for a product of functions Derivative for a quotient of functions Derivative for a composite of function

Examples 1. If y = 2x 2 4x + 1 60, find D x y. 13 t 2. Find D 3 2t+1 t. t 4 +3 3. Find F (y), where F y = y sin y 2. 4. Find D x sin cos x 2.

2.6 Higher Order Derivatives

Notations for derivatives Example. If y = sin 2x, find d 12 y dx 12

Implicit Differentiation 1. Find dy/dx if 4x 2 y 3y = x 3 1. 2. If s 2 t + t 3 = 1, find ds/dt and dt/ds. 3. Sketch the graph of the circle x 2 + 4x + y 2 + 3 = 0 and then find equations of the two tangent lines that pass through the origin.

Related Rates 1. Each edge of a variable cube is increasing at a rate of 3 inches per second. How fast is the volume of the cube increasing when an edge is 12 inches long? 2. Water is pouring into a conical tank at the rate of 8 cubic feet per minute. If the height of the tank is 12 feet and the radius of its circular opening is 6 feet, how fast is the water level rising when the water is 4 feet deep? 3. An airplane flying north at 640 miles per hour passes over a certain town at noon. A second airplane going east at 600 miles per hour is directly over the same town 15 minutes later. If the airplanes are flying at the same altitude, how fast will they be separating at 1:15 PM?

2.9 Differential and Approximation

y and dy If x small then y the actual change in y dy an approximation to y

Differentials

Derivative vs Differential

Approximations Examples. 1. Suppose you need a good approximations to 4.6 and 8.2, but your calculator has died. What might you do? 2. Use differentials to approximate the increase in the area of a soap bubble when its radius increases from 3 inches to 3.025 inches.

Estimating Errors 1. The side of a cube is measured as 11.4 centimeters with a possible error of ±0.05 centimeter. Evaluate the volume of the cube and give an estimate for the possible error in this value. absolute error vs relative error 2. Poiseuille s Law for blood flow says that the volume flowing through an artery is proportional to the fourth power of the radius, that is, v = kr 4. By how much must the radius be increased in order to increase the blood flow by 50%?

Linear Approximation L x = f a + f (a)(x a) Find and plot the linear approximation to f x = 1 + sin 2x.