1. Calculus and Analytic Geometry I Cloud County Community College Fall, 2012 Instructor: Timothy L. Warkentin

2. Chapter 03: Differentiation 03.01 Tangents and the Derivative at a Point 03.02 The Derivative as a Function 03.03 Differentiation Rules 03.04 The Derivative as a Rate of Change 03.05 Derivatives of Trigonometric Functions 03.06 The Chain Rule 03.07 Implicit Differentiation 03.08 Derivatives of Inverse Functions and Logarithms 03.09 Inverse Trigonometric Functions 03.10 Related Rates 03.11 Linearization and Differentials

3. Chapter 03 Overview The logic of chapter 03: –Definition of the slope of a curve at a point. –Definition of the line tangent to a curve at a point. –Definition of the derivative of a function at a point. –Definition of the derivative of a function. –Rules for finding derivatives of functions. –Meanings of the derivative. –Applications of the derivative.

4. 03.01: Tangents and the Derivative at a Point 1 The slope of a curve at a point is the slope of the limiting secant line as h shrinks to zero. Lab: Slope of Curve

5. 03.01: Tangents and the Derivative at a Point 2 The slope of the curve y = f [x] at the point (x 0, y 0 ) is the limit of the difference quotient. Examples 1 & 2 The tangent line to the curve y = f [x] at (x 0, y 0 ) is the line through (x 0, y 0 ) with a slope equal to the slope of the curve at (x 0, y 0 ). Vertical tangent lines have infinite slopes. The derivative of the function y = f [x] at the point (x 0, y 0 ). Example 2

6. 03.02: The Derivative as a Function 1 The derivative of the function y = f [x] is a function derived from f [x] that gives the slope of the curve at any point (x, f [x]). Precise notation for the derivative of a function. Ambiguous (but very common) notation for the derivative of a function.

7. 03.02: The Derivative as a Function 2 There are two equivalent forms for the definition of the derivative of a function. One form may be substantially easier to use on any particular problem. Examples 1 & 2 The relationship between the graphs of a function and its derivative. Example 3, Derivative Puzzles 1, 2, & 3

8. 03.02: The Derivative as a Function 3 A function is Differentiable at point (x 0, y 0 ) if is a real number. Example 4 Differentiability on open and closed intervals. Example 5 A function will not be differentiable wherever its graph exhibits a corner, a cusp, a vertical tangent or a discontinuity (section 02.05). Diagrams – page 130 The Intermediate Value Property of Derivatives (Darboux’s Theorem): The Intermediate Value Theorem for Continuous Functions holds for the derivative of a differentiable function.

9. 03.02: The Derivative as a Function 4 It can be shown that Thus but

10. 03.03: Differentiation Rules 1 Constant Rule. Power Rule. Example 1 Constant Multiple Rule. Example 2 Sum/Difference Rule. Examples 3 & 4 Natural Base Rule. The function y = e x is its own derivative. The height of the of y = e x curve is the slope of the curve. Example 5 Product Rule. Examples 6 & 7 Quotient Rule. Examples 8 & 9 Higher Order Derivatives. Example 10

11. 03.04: The Derivative as a Rate of Change 1 The Instantaneous Rate of Change (IRC) of f with respect to x is Example 1 Given that x[t] is the position of a particle at time t as it moves along the x-axis then Example 2

12. 03.04: The Derivative as a Rate of Change 2 The vertical position of a body under the influence of gravity is where g is the gravitational constant. Examples 3 & 4

13. 03.05: Derivatives of Trigonometric Functions 1 The six trigonometric derivatives should be memorized. Examples 1 – 7

14. 03.06: The Chain Rule 1 The chain rule is used to find the derivative of composed functions. The power of the chain rule along with the other derivative rules allows the differentiation of nearly every useful function. Examples 1 – 9

15. 03.07: Implicit Differentiation 1 Since the variable y is assumed to be a function of x, the chain rule must be used to take the derivative of every term that contains the variable y. Examples 1 – 5 (use Graphing Calculator to graph implicit functions)

16. 03.08: Derivatives of Inverse Functions and Logarithms 1 If point (a,b) is on f [x] then the point (b,a) is on f -1 [x]. If f [a] = b then the inverse statement is a = f -1 [b]. The derivative of the inverse function at the point (b,a) is. This relationship allows the calculation of the numerical value of the derivative of an inverse function without actually knowing the derivative of the inverse function explicitly. It does however require knowing both coordinates of the point involved. This means the problem b = f [a] must be solved for ‘a’. Examples 1 & 2

17. 03.08: Derivatives of Inverse Functions and Logarithms 2 Parametrizing inverse functions. Derivation of y = ln [x]. Examples 3 & 4 Why is there an absolute value in ? The derivative of y = a x. Example 5 The Change of Base formula. The derivative of y = log a x. #77, Exercises 3.8 Logarithmic Differentiation. Examples 6 & 7 The Number e as a limit.

18. 03.09: Inverse Trigonometric Functions 1 The six trigonometric functions (sine, cosine, tangent, cotangent, secant, and cosecant) do not have inverses because they are not 1-to-1 functions. Creating six new trigonometric functions (Sine, Cosine, Tangent, Cotangent, Secant, and Cosecant) that are 1- to-1 by doing domain surgery. Defining. Example 1 Using inverse statements. #6, Exercises 3.9 Finding the derivative of inverse Sine using the rule for the derivative of an inverse function from section 03.08. Finding the derivative of inverse Sine using a triangle and implicit differentiation.

19. 03.09: Inverse Trigonometric Functions 2 Derivatives of the inverse trigonometric functions. Examples 2 & 3

20. 03.09: Inverse Trigonometric Functions 3 Inverse Function – Inverse Function Identities

21. 03.10: Related Rates 1 The problem of finding a rate that cannot easily be measured from a rate that can be measured directly is called a related rates problem. A relationship between the variables involved in the problem is obtained from geometric or algebraic considerations and then this relationship is implicitly differentiated with respect to time. Related Rates Problem Strategy (page 194). Examples 1 – 6

22. 03.11: Linearization and Differentials 1 Linearization is the process of approximating a small segment of a curve with its tangent line. Examples 1 – 3

23. 03.11: Linearization and Differentials 2 The differential of function is defined as the product. Examples 4 & 5 Confusion results when the shorthand symbol is used for the derivative and it appears to be a ratio of differentials. The derivative should be (but is usually not) correctly written as.

24. 03.11: Linearization and Differentials 3 Estimating with differentials (differential change verses the true change). Examples 6 & 7 The availability of computing machines has lessened the value of estimation methods.