CHAPTER 3 DIFFERENTIATION

Transcription

1 CHAPTER 3 DIFFERENTIATION

2 3.1 THE DERIVATIVE AND THE TANGENT LINE PROBLEM You will be able to: - Find the slope of the tangent line to a curve at a point - Use the limit definition to find the derivative of a function - Understand the relationship between differentiability and continuity

3 TANGENT LINE PROBLEM Find the tangent line to a curve Finding the slope of the tangent line at point P Approximate using a secant line through the point of tangency and a second point

4 DEFINITION OF TANGENT LINE WITH SLOPE M As the change in x approaches 0, the difference quotient gets closer and closer to the slope of the line at the point of tangency, hence the definition of the slope of the tangent line: This limit is actually one of the two fundamental operations in calculus - DIFFERENTIATION

5 DIFFERENTIATION NOTATION All of these symbols refer back to differentiation Most common are the first 3 this symbol is read as prime so f prime of x dy/dx is read the derivative of y with respect to x

6 DIFFERENTIABILITY AND CONTINUITY The derivative of f at c is This is provided the limit exists, which means that both sides of the limit agree, hence continuity If f is differentiable at x=c, then f is continuous at x=c

7 HOMEWORK

8 3.2 BASIC DIFFERENTIATION RULES AND RATES OF CHANGE You will be able to: - Find the derivative of a function using the Constant Rule, Power Rule, Constant Multiple Rule, Sum and Difference Rules - Find the derivative of the sine, cosine, and exponential functions - Use derivative to find rates of change

9 CONSTANT AND POWER The Constant Rule The derivative of a constant function is 0. If c is a real number then The Power Rule If n is a rational number, then the function differentiable and is

10 CONSTANT MULTIPLE, SUM, AND DIFFERENCE If f is a differentiable function and c is a real number, then cf is also differentiable and The sum (or difference) of two differentiable functions f and g is itself differentiable. Moreover, the derivative of f+g (or f-g) is the sum (or difference) of the derivatives of f and g

11 SINE, COSINE, AND E

12 RATES OF CHANGE Distance = Rate x Time, Rate = distance/time s is the position function of time t Average velocity is change in distance/change in time = Instantaneous velocity is the slope of the position function Position function of a free-falling object:

13 HOMEWORK

14 3.3 PRODUCT AND QUOTIENT RULES AND HIGHER ORDER You will be able to: - find the derivative of a function using the Product and Quotient Rules - find the derivative of a trigonometric function - find a higher order derivative of a function

15 THE PRODUCT RULE The product of two differentiable functions f and g is itself differentiable Words: The derivative fg is the first function times the derivative of the second function, plus the second function times the derivative of the first function Symbols: Mnemonic 1 d2 + 2 d1

16 THE QUOTIENT RULE The quotient f/g of two differentiable functions f and g is itself differentiable at all values of x for which g(x) is not 0. Words: The derivative of f/g is given by the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator. Symbols: Mnemonic: Low d-high minus high d-low over the square of whats below

17 DERIVATIVES OF TRIGONOMETRICS Something to remember: sec and tan go together, cot and csc go together and are negative

18 HIGHER ORDER DERIVATIVES The second derivative is simply the derivative of the derivative Mainly used since acceleration is the slope of the velocity or the second derivative of the position function: a(t)=v (t)=s (t)

19 HOMEWORK

20 3.4 THE CHAIN RULE You will be able to: - find derivatives using the Chain Rule and General Power Rule

21 THE CHAIN RULE If y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x, then y = f(g(x)) is a differentiable function of x and or equivalently: For the chain rule, identify the inside function and the outside function

22 THE GENERAL POWER RULE AND NATURAL LOG If, where u is a differentiable function of x and n is a rational number, then Let u be a differentiable function of x.

23 ABSOLUTE, DIFFERENT BASES If u is a differentiable function of x such that u is not 0, then Derivatives of Bases other than e If a is a positive real number (a is not 1) and x is any real number then If a is a positive real number (a is not 1) and x is any positive real number then

24 BOOKMARK P. 160 HOMEWORK

25 3.5 IMPLICIT DIFFERENTIATION You will be able to: - distinguish between functions written in implicit form and explicit form - use implicit differentiation to find the derivative of a function - find derivatives of functions using logarithmic differentiation

26 IMPLICIT VS EXPLICIT Implicit: xy = 1 Explicit: y=1/x Differentiation with respect to x (dy/dx) (dx/dx)

27 IMPLICIT DIFFERENTIATION Guidelines Differentiate both sides with respect to x Collect all terms involving dy/dx on the left side of the equation and move all other terms to the right side of the equation Factor dy/dx out of the left side of the equation Solve for dy/dx by dividing both sides of the equation by the left-hand factor that does not contain dy/dx

28 LOGARITHMIC DIFFERENTIATION On occasion, it is convenient to use logarithms as aids in differentiation nonlogarithmic functions This means we take the ln of both sides and then derive

29 HOMEWORK

30 3.6 DERIVATIVES OF INVERSE FUNCTIONS You will be able to: - Find the derivative of an inverse functions - Differentiate an inverse trigonometric function - Review that basic differentiation formulas for elementary functions

31 INVERSE FUNCTIONS Let f be a function whose domains is an interval I. If f has an inverse function, then the following statements are true. If f is continuous on its domain, then the inverse of f is continuous on its domain If f is differentiable on an interval containing c and f (c) is not 0, then the inverse of f is differentiable at f(c) If f has an inverse function g then

32 DERIVATIVES OF INVERSE TRIG FUNCTIONS Let u be a differentiable function of x.

33 HOMEWORK

34 3.7 RELATED RATES You will be able to: - Find a related rate - Use related rates to solve real-life problems

35 GUIDELINES FOR SOLVING RELATED-RATE PROBLEMS Identify all given quantities and quantities to be determined. Make a sketch and label the quantities. Write an equation involving the variables whose rates of change either are given or are to be determined. Using the Chain Rule, implicitly differentiate both sides of the equation with respect to time t. After completing Step 3, substitute into the resulting equation all known values for the variables and their rates of change. Then solve for the required rate of change.

36 HOMEWORK

37 3.8 NEWTON S METHOD You will be able to approximate a zero of a function using Newton s Method.

38 NEWTON S METHOD Basically Newton s Method is using tangent lines to approximate the graph of the function near its x-intercepts. Let f(c)=0, where f is differentiable on an open interval containing c. To approximate c, use the following steps: Make an initial estimate that is close to c (graph) Determine a new approximation: If is within the desired accuracy, let serve as the final approximation. Otherwise return to Step 2 and calculate again.

39 HOMEWORK