1. AP Calculus AB Review Part One

2. Limits  You should be able to  Understand limits, including asymptotic and unbounded behavior  Calculate limits using algebra  Estimate limits from graphs and data  Understand one-sided limits  Understand continuity in terms of limits

3. Limit Strategies  Always begin with direct substitution  Try to simplify or cancel  Try to rationalize  Use table or graph on your calculator  Two Special Limits  The limit as x approaches 0 of (sinx/x) = 1  The limit as x approaches 0 of (1-cosx / x) = 0

4. THE LIMIT LAWS

5. Consider: If we try to evaluate this by direct substitution, we get: In this case, we can evaluate this limit by factoring and canceling: Zero divided by zero can not be evaluated, and is an example of indeterminate form.

6. Use the Limit Laws and the graphs of f and g in the figure to evaluate the following limits, if they exist. a. b. c. USING THE LIMIT LAWS

7. From the graphs, we see that and.  Therefore, we have: USING THE LIMIT LAWS

8. We see that. However, does not exist—because the left and right limits are different: and USING THE LIMIT LAWS Example 1 b

9. However, we can use the Product Law for the one-sided limits: and  The left and right limits aren ’ t equal.  So, does not exist. USING THE LIMIT LAWS Example 1 b

10. The graphs show that and. As the limit of the denominator is 0, we can ’ t use the Quotient Law.  does not exist.  This is because the denominator approaches 0 while the numerator approaches a nonzero number. USING THE LIMIT LAWS Example 1 c

11. Find where.  Here, g is defined at x = 1 and.  However, the value of a limit as x approaches 1 does not depend on the value of the function at 1.  Since g ( x ) = x + 1 for, we have: USING THE LIMIT LAWS Example 4

12. Find  If we begin by plugging in our limit, we find that the limit of the denominator is 0.  We must use algebra to find the limit  Rationalizing USING THE LIMIT LAWS

13.  Thus,

14. Show that  Recall that:  Since | x | = x for x > 0, we have:  Since | x | = - x for x < 0, we have: USING THE LIMIT LAWS

15. The result looks plausible from the figure. USING THE LIMIT LAWS

16. Prove that does not exist.  Since the right- and left-hand limits are different, it follows that does not exist. USING THE LIMIT LAWS

17. The graph of the function is shown in the figure. It supports the one-sided limits that we found. USING THE LIMIT LAWS

18. If determine whether exists.  Since for x > 4, we have:  Since f ( x ) = 8 - 2 x for x < 4, we have: USING THE LIMIT LAWS Example 9

19.  The right- and left-hand limits are equal.  Thus, the limit exists and. USING THE LIMIT LAWS Example 9

20. The graph of f is shown in the figure. USING THE LIMIT LAWS Example 9

21. L ’ Hôpital ’ s Rule: If is indeterminate, then:

22. Example

23. On the other hand, you can apply L ’ Hôpital ’ s rule as many times as necessary as long as the fraction is still indeterminate: not (Rewritten in exponential form.)

24. Example: If it ’ s no longer indeterminate, then STOP ! If we try to continue with L ’ Hôpital ’ s rule: which is wrong, wrong, wrong!

25. Limits approaching infinity (HA)  To find a limit as x approaches infinity (or negative infinity), express the function as a fraction  BOBOBOTNEATSDC (degree check)  Bigger on bottom zero  Bigger on top none  Each are the same, divide coefficients  When the numerator or denominator does not contain polynomials, try to see the effect of plugging in very large (or very small numbers)  Divide out your variable of the largest degree

26. We can use limits to find the derivative of a function  The derivative is the slope of the tangent line and can be found by using the limit:  The derivative at a point can be found using a limit with the “slope equation”

27. Tangent Definition  From geometry  a line in the plane of a circle  intersects in exactly one point  We wish to enlarge on the idea to include tangency to any function, f(x)

28. Slope of Line Tangent to a Curve  Approximated by secants  two points of intersection  Let second point get closer and closer to desired point of tangency

29. Animated Tangent

30. Slope of Line Tangent to a Curve  Use the limit in this context

31. Definition of a Tangent Let Δ x shrink from the left

32. Definition of a Tangent  Let Δx shrink from the right

33. The Slope Is a Limit  Consider f(x) = x 3 Find the tangent at x 0 = 2  Now finish …

34. Animated Secant Line

35. SLOPES  Determining the slope of the line tangent to a specific point x = c is not very difficult. After getting the derivative, you plug in the point. For example, find the slope of the line tangent to x = 3 for the graph y = x 2.  Find the derivative. The derivative is 2x.  Then you put 3 in for x. 2(3) = 6.  So the slope of the line tangent to x=3 for y = x 2 is 6.

36. TANGENT LINES  Find the equation for the line tangent to x = 3 for y = x 2.  First you need to know the (x,y) points for x=3. If x = 3, then y = 9. We also know that the slope at x =3 is 6. So at point (3,9), the slope is 6. Remember the equation for a line?

37. TANGENT LINES  m = 6 (derivative of f(x) at x = 3.)  x = 3 (given)  y = 9 (f(3) is 9.)  x 0 and y 0 are the points on the graph.  m is the slope. x and y are the variables. OR

38. DIFFERENTIABILITY  A function is known to be differentiable if you are able to take derivatives of the function within the range [a,b]. The following four are cases where the function is not differentiable at certain points. CORNERSCUSPS DISCONTINUITIES VERTICAL TANGENTS

39. LOGIC  Therefore, if a function is differentiable, then it is continuous.  However, if a function is continuous, it is not always differentiable. (y=|x| has a corner which is not differentiable at x=0.)

40. DIFFERENTIATION RULES  y,u and v are functions of x. a,b,c, and n are constants (numbers). The derivative of a constant is zero. Duh! If everything is constant, that means its rate, its derivative, will be zero. The graph of a constant, a number is a horizontal line. y=c. The slope is zero. The derivative of x is 1. Yes. The graph of x is a line. The slope of y = x is 1. If the graph of y = cx, then the slope, the derivative is c.

41. MORE RULES  When you take the derivative of x raised to a power (integer or fractional), you multiply expression by the exponent and subtract one from the exponent to form the new exponent.

42. OPERATIONS OF DERIVATIVES  The derivative of the sum or difference of the functions is merely the derivative of the first plus/minus the derivative of the second. The derivative of a product is simply the first times the derivative of the second plus second times the derivative of the first. The derivative of a quotient is the bottom times the derivative of the top, minus top times the derivative of the bottom….. All over bottom square.. TRICK: LO-DEHI – HI- DELO LO 2

43. JUST GENERAL RULES  If you have constant multiplying a function, then the derivative is the constant times the derivative. See example below:  The coefficient of the x 6 term is 5 (original constant) times 7 (power rule.)

44. SECOND DERIVATIVES  You can take derivatives of the derivative. Given function f(x), the first derivative is f ’ (x). The second derivative is f ’’ (x), and so on and so forth.  Using Leibniz notation of dy/dx

45. EXAMPLE:  Find the derivative:  Use the power rule and the rule of adding derivatives.  Note 3/2 – 1 = ½. x ½ is the square root of x.

46. EXAMPLE 5  Find the equation of the line tangent to y = x 3 +5x 2 –x + 3 at x=0.  First find the (x,y) coordinates when x = 0. When you plug 0 in for x, you will see that y = 3. (0,3) is the point at x=0.  Now, get the derivative of the function. Notice how the power rule works. Notice the addition and subtraction of derivative. Notice that the derivative of x is 1, and the derivative of 3, a constant, is zero.

47. EX 5 (continued)  Now find the slope at x=0, by plugging in 0 for the x in the derivative expression. The slope is -1 since f ’ (0) = -1.  Now apply it to the equation of a line.

48. EX 5. (continued)  Now, plug the x and y coordinate for x 0 and y 0 respectively. Plug the slope found in for m.  And simplify  On the AP, you can leave your answer as the first form. (point-slope form)

49. EXAMPLE 6  Find all the derivatives of y = 8x 5.  Just use the power rule over and over again until you get the derivative to be zero.

50. TRIG DERIVATIVES  In addition to those rules, you will also need to know how to get derivatives of the six trigonometric functions.  Getting the derivatives for any function starts from the definition of the derivative. However, deriving the six trig functions using this is very tedious and not even practical to discuss it in this course. Therefore you will merely need to memorize all six.  If you are pretty good at trigonometry, you will know that sine and cosine functions are the only functions from which tangent, secant, cosecant and cotangent come from. To derive tangent, secant, cotangent, and cosecant, we will use their previous theorems we have learned, plus the derivative of the sine and cosine.

51. DERIVATIVE OF SINE AND COSINE  After a very tedious and rigorous process of using the difference quotient and definition of derivative, you will get the derivative of sine and cosine. Not hard. Just be careful when you are positive or negative.

52. DERIVATIVE OF TANGENT  You can find the derivative of tan x by knowing that tan x is no different than…  (sin x)/(cos x).  Thus, we can use the quotient rule to find the derivative of tan x. sin 2 x+cos 2 x=1 TRIG IDENTITY

53. DERIVATIVE OF RECIPROCAL FUNCTIONS  You can find the derivative of the reciprocal functions (cosecant, secant, and cotangent.) Just know that the secant is the reciprocal of the cosine, the cosecant is the reciprocal of the sine and the cotangent is the reciprocal of tangent.  You could use the quotient rule by sec x = 1/(cos x), csc x = 1/(sin x), and cot x = 1/(tan x).

54. DERIVATION OF DERIVATIVES OF THE RECIPROCAL FUNCTIONS

55. COTANGENT  Cotangent is done easier using the identity that…  cot x = (cos x)/(sin x)

56. PRACTICE PROBLEMS  GIVEN  PRODUCT RULE  d(cot x) = -csc 2 x  d(x)=1  Simplified Find the derivative of: x (cot x)

57. Another example:  Find the derivative of:  GIVEN  QUOTIENT RULE  d(xsinx) use of PRODUCT RULE

58. CHAIN RULE  Say for example, you are given a function, f(x) = (x+1) 2. You are asked to find the derivative.  You could FOIL it and get x 2 +2x+1 and differentiate term by term and get 2x+2.  Say you had (x+1) 5 and you wanted to find the derivative. Tedious, but you could use the binomial theorem and get x 5 +5x 4 +10x 3 +10x 2 +5x+1 and differentiate term by term to get 5x 4 +20x 3 +30x 2 +20x+5.  Try to differentiate sin(5x). There is no real identity to help you with that, so you can ’ t rename this function in order to simplify it. The argument (the thing inside the parenthesis) is forbidding us to use the typical rules. How are we able to differentiate?

59. THE CHAIN RULE  The chain rule says that in order to take the derivative of a composition of functions f(g(x)), you differentiate f first, and multiply f ’(g) with g ’.  Looks like this. Let h(x) be a composite differentiable function f(g(x)).  h ’ (x) = f ’ (g(x)) g ’ (x)

60. CHAIN RULE (u-sub)  For y=(2x-3) -2 say that you let 2x-3 = u. you would get u - 2.  Take the derivative of u -2. You will get -2u -3.  Multiply that with the derivative of u. The derivative of u is 2.  So your answer will be -4(2x-3) -3.

61. CHAIN RULE FORMULA  Given that u is a differentiable function of x, and y is a composition function with u and x, then..

62. EXAMPLE  Find the derivative of y=2sin(2x 2 -3)  Given and declaring u.  y(u): y is written in terms of u.  CHAIN RULE  dy/du and du/dx found  Replacing u and simplifying.

63. CHAIN RULE  Most of the time, your argument will be u. If you have a function y(x).  Then the derivative of y(x) is the derivative of y with respect to u (the argument) multiplied with the derivative of u (the argument) with respect to x.  You can also have three functions: k(f(g (x))) The derivative would be k’(f(g(x)))f’(g(x))g’(x)

64. IMPLICIT DIFFERENTIATION  A function y(x) which is solved for y in terms of x is called explicit functions. y=sin x is an example of an explicit function since it is solved for y and the function is only in terms of x.  However, there will be relations which are not functions. They will have a combination of x and y. For example: x 2 -3y-8=y 2. These functions with both x and y together is considered an implicit function. Consider that y is a hidden ( implicit ly defined) function of x.

65. IMPLICIT DIFFERENTIATION.  The process of finding derivatives, the slope of the line tangent to a point, of an implicitly defined relation is called implicit differentiation.  Remember that dy/dx really means “ Derivative of y (dy) with respect to x (dx). ”  The derivative of 2x with respect to x is 2.  But the derivative with respect to y is dy/dx.  The derivative with the respect to y 2 is (2y)(dy/dx). Remember, y is a hidden function of x. We are differentiating with respect to x, not y.  Remember chain rule? The derivative of u 2 with respect to x is 2u (du/dx).

66. EXAMPLE:  Differentiate x 2 +y 2 =1  Sum of derivatives  Differentiate.. y is a function of x, so you have to do chain rule  Solve for dy/dx algebraically.  Yes, its okay to have a y in the derivative.

67. IMPLICIT DIFFERENTIATION  You could have solve the problem by solving for y.  Using the chain rule have 1- x 2 be u, you can find the derivative.  Since the entire denominator is equal to y, you can replace the denominator with y.

68. Sum of Differentiation Rules All of the rules you forgot in one(ish) place ---