Limits, Continuity and the Derivative

1. Limits, Continuity and the Derivative

2. The Derivative and the Slope of a Graph Consider the case of an object moving along a curve from left to right. At the point where the object is released, it continues in a straight line. This line is called the tangent line. The slope of the tangent line is the direction of the object at that instant. So we can think of the slope of a curve as the slope of the tangent line to the curve at a given point.

3. The Derivative and the Slope of a Graph Unlike a straight line, a curve changes direction. So the slope of a curve changes from one point to another. Thus we speak of the slope of the curve at a point. The slope of the tangent line to the curve at a point is called the Derivative of the curve at that point.

4. Using Limits to calculate the derivative Recall that the derivative at the indicated point is the slope of the tangent line (indicated here) at that point. We usually need two points on the tangent line to determine its slope. However, we only have one point (indicated). How can we determine the slope of the tangent line using only one point?

5. Using Limits to calculate the derivative An approximate value for the slope of the tangent line can be obtained from the slope of a secant line that passes through the given point. The secant line cuts the curve at points (a, f(a)) and (a+h, f(a+h)). Note that the difference between the x-values is h. The slope of the secant line is therefore: This is called the Difference Quotient.

6. Using Limits to calculate the derivative A better approximation to the slope of the tangent line will be achieved if we choose a point closer to the given point. That is, if we make h smaller. Note that slope of the cyan secant line is closer to the slope of the tangent line, than the original green secant line that we chose. The value of h is smaller for the cyan secant line than for the original green secant line.

7. Using Limits to calculate the derivative As the second point gets closer to the given point (that is, as h approaches zero), then the slope of the secant line approaches the slope of the tangent line. This gives rise to the following limit formula for the slope of the tangent line at the point (a, f(a)): This is called the Derivative of f(x) at the point where x=a, or f ’(a). We say “f -prime of a.”

8. Using Limits to calculate the derivative So the derivative of f(x) at the point (a, f(a)) is: IMPORTANT! Do not forget that: The derivative of a function at a point is the slope of the tangent line to the curve at the given point.

9. Example: Find the equation of the tangent line to f(x) at the point (2, 1) if Solution: To find the equation of the tangent line, we need the slope of the line and a point. The point is (2, 1). The slope of the tangent line is f ‘(2), since x = 2 at the point of tangency. Using the limit definition of the derivative: ; Continued on the next slide…

10. Example (Cont’d): Find the equation of the tangent line to f(x) at the point (2, 1) if Solution (Cont’d): Note that both the numerator & denominator equal 0 when h = 0 is substituted. So we must simplify the fraction and divide numerator & denominator by h (that is, cancel h). Then substitute for h and simplify. 1 1 Continued on the next slide…

11. Example (Cont’d): Find the equation of the tangent line to f(x) at the point (2, 1) if Solution (Cont’d): 1 1 So the slope of tangent line m = – 1. Equation of tangent line through point (2, 1) and slope m = – 1 is:

12. The Derivative Function The derivative function (or just the derivative) is a function that can provide the slope of the tangent line for a given x-value: Example:The derivative function for So the derivative at point (3, 9) is Check if this is so using the limit definition of the derivative at x = 3. That is, determine

13. The Derivative Function To calculate the derivative function, simply replace a with x in the limit definition of derivative, then solve for a function of x. That is, calculate: Example:Determine the derivative of Solution: Continued on the next slide…

14. Example (Cont’d): Find for To cancel h, rationalize the numerator. Solution (Cont’d): Ans.: So Eliminate the lim notation when you substitute h = 0. The notation remains until the substitution occurs.

15. Example (Cont’d): So, if then Now it is easy to determine the slopes for various x-values: ; Note that, even though f(0) = 2 (that is, 0 is in the domain of f ), (Does Not Exist) Why?

16. Continuity & The Derivative Since the derivative is a limit (as h→ 0), and since the point of tangency must be on the curve, then A function does not have a derivative where it is not continuous. In other words… If a function is differentiable at x = c, then the function must be continuous at x = c. This means: • If a function is differentiable (has a derivative) at a point or over an interval, then it is continuous at that point or on that interval.

17. Continuity & The Derivative Important Note: Although all differentiable functions are continuous, Not all continuous functions are differentiable! This means: • It only works one direction: IF Differentiable, THEN Continuous Not necessarily vice versa.

18. Notation for Differentiation One notation for derivative is already familiar: The derivative of f(x) is denoted f ‘(x). Another notation is called Leibniz notation: The derivative of y with respect to x is So Other notations for the derivative (with respect to x) of a function y = f(x): We will use primarily the first three of notations.

19. Some Rules of Differentiation Question: What is the tangent line to a straight line? Answer: The line itself. That is, a straight line is its own tangent line. Since the derivative is the slope of the tangent line, then the derivative of a straight line (at any point) is the slope of the line: In particular, Based on this, since a constant is a horizontal straight line, then the derivative of a constant is zero:

20. Some Rules of Differentiation Question: What is the tangent line to a straight line? Answer: The line itself. That is, a straight line is its own tangent line. Since the derivative is the slope of the tangent line, then the derivative of a straight line (at any point) is the slope of the line: In particular, Based on this, since a constant is a horizontal straight line, then the derivative of a constant is zero: Continued on the next slide…

21. Some Rules of Differentiation Examples: Derivative of a constant is 0. Continue for more rules…

22. Some Rules of Differentiation Simple Power Rule: n is any real number. Example: Constant Multiple Rule: c is a constant. Example: Example: Continue for more rules…

23. Some Rules of Differentiation Sum and Difference Rules: “Derivative of a sum is the sum of the derivatives.” “Derivative of a difference is the difference of the derivatives.” Example: Continue for more rules…

24. Some Rules of Differentiation Product Rule: In abbreviated form (which may be easier to remember): Read: “Derivative of f times g equals f-prime g plus f g-prime.” Example:Differentiate Step 1:Identify f(x) and g(x) Step 2:Determine f‘(x) and g’(x) Continued on the next slide…

25. Some Rules of Differentiation Product Rule Example (Cont’d): Step 3:Substitute the f(x), g(x), f‘(x) and g’(x) into the Product Rule formula, and simplify: Continue for more rules…

26. Some Rules of Differentiation Quotient Rule: In abbreviated form (which may be easier to remember): Read: “Derivative of f over g equals f-prime g minus f g-prime divided by g squared.” Example:Differentiate Step 1:Identify f(x) and g(x) Step 2:Determine f‘(x) and g’(x) Continued on the next slide…

27. Some Rules of Differentiation Quotient Rule Example (Cont’d): Step 3:Substitute the f(x), g(x), f‘(x) and g’(x) into the Quotient Rule formula, and simplify: Continue for more rules…

28. Some Rules of Differentiation Chain Rule: The Chain Rule is the differentiation of composition of functions. Recall that a composition of functions is a “function of a function.” The output of one function (the “inner” function) is the input of the other function (the “outer” function), as indicated below. x Continued on the next slide…

29. Some Rules of Differentiation Chain Rule: “The derivative of a composition of functions is equal to the derivative of the outer function with respect to the inner function (that is, without changing the inner) multiplied by the derivative of the inner function.” Outer function Example: Derivative of inside Derivative of outside

30. Some Rules of Differentiation Additional Chain Rule Notation: Example:

31. Some Rules of Differentiation Special Case of the Chain Rule – The General Power Rule: Example:Differentiate Solution:

32. Higher-Order Derivatives Technically, what we have been referring to as the derivative is actually the first derivative. That is, it is the function obtained when we differentiate a function once. If we differentiate again, the result is called the second derivative. That is, the second derivative is the derivative of the first derivative. Subsequent derivatives are named similarly. For example, the seventh derivative of a function is obtained by taking derivatives seven times (the derivative of the derivative of the derivative of the derivative of the derivative of the derivative of the derivative of the function).

33. Higher Order Derivatives Notation: First Derivative: Second Derivative: Third Derivative: Fourth Derivative:

34. Higher Order Derivatives Example: Solution:

35. Implicit Differentiation Recall the Chain Rule: Recall also that the Chain Rule applies to a composition of functions: f(g(x)). Now, examine the following example: Ans.: …using the Chain Rule… Continued on the next slide…

36. Implicit Differentiation Example continued: …using the Chain Rule: So… Note: • The derivative resembles a regular derivative • An additional dy/dx is multiplied in the derivative. Why? • We use the Chain Rule to do the derivative.

37. Implicit Differentiation A Loose Description of Implicit Differentiation: To differentiate a function of y with respect to x: • Differentiate the function as usual (in terms of y), then • Multiply by dy/dx . Note:When differentiating keep the following in mind: • Always differentiate BOTH SIDES of the equation with respect to the same variable. • The variable that we differentiate with respect to occurs in the denominator of the derivative expression. For example, if we are seeking dy/dx, then differentiate with respect to x. If we are seeking dV/dt, then differentiate with respect to t.

38. Implicit Differentiation Example: Steps: • Differentiate both sides with respect to x. Use the sum/difference rule where necessary. • Determine whether the term we differentiate contains x or y. If it is a function of x, then regular derivatives (since we differentiate with respect to x). If it is a function of a variable other than x, (y in this case), then it is implicit differentiation. Note that we differentiate both sides with respect to x. This term is a function of x, so regular differentiation. This term is a function of y, so Implicit differentiation.

39. Implicit Differentiation Example (Cont’d): Steps: • Differentiate each term using the appropriate rules of differentiation. Remember, for implicit differentiation, differentiate as usual, but multiply by dy/dx at the end. Derivative of a constant is 0 Regular differen-tiation Implicit differen-tiation • Solve the equation for dy/dx.

40. Implicit Differentiation Another Example: Steps: • Differentiate both sides with respect to x. Use the sum/difference rule where necessary. • To differentiate a product, use the Product Rule. Be sure to put all x in one fxn and all y in the other. One fxn in terms of x One fxn in terms of y Continued on next slide…

41. Implicit Differentiation Example 1 (Cont’d): Steps: • Note that both parts of the product are in fxns of x: f(x) & g(x). • When doing each differentiation, be sure to identify whether you need to do implicit differentiation or regular differentiation. Regular differentiation, since f(x) is a fxn of x and we differentiate dx. Implicit differentiation, since f(x) is a fxn of y and we differentiate dx. Continued on next slide…

42. Implicit Differentiation Example 1 (Cont’d): Steps: • Complete the Product Rule. Be careful to substitute carefully. • Do the same for all products. Implicit differentiation, since f(x) is a fxn of y and we differentiate dx. Regular differentiation, since g(x) is a fxn of x and we differentiate dx. Continued on next slide…

43. Implicit Differentiation Example 1 (Cont’d): Steps: • Substitute all derivatives into the original equation. • Since we wish to find dy/dx at point (2, –1), substitute x = 2 & y = –1, then solve for dy/dx. Answer

44. For more examples on how to find Derivatives, including Implicit Differentiation, check the following websites: http://www.sosmath.com/calculus/diff/der05/der05.html and http://archives.math.utk.edu/visual.calculus/2/index.html