Review of elements of Calculus (functions in one variable)

Transcription

1 Review of elements of Calculus (functions in one variable) Mainly adapted from the lectures of prof Greg Kelly Hanford High School, Richland Washington

2 Functions Golden Gate Bridge San Francisco, CA Photo by Vickie Kelly, 004 Greg Kelly, Hanford High School, Richland, Washington

3 A relation is a function if: for each there is one and only one y. A relation is a one-to-one if also: for each y there is one and only one. In other words, a function is one-to-one on domain D if: f a f b whenever a b

4 To be one-to-one, a function must pass the horizontal line test as well as the vertical line test y y y one-to-one not one-to-one not a function (also not one-to-one)

5 Inverse functions: f y Solve for : y y y Switch and y: y Given an value, we can find a y value f Inverse functions are reflections about y =.

6 Inverse of function f a This is a one-to-one function, therefore it has an inverse. Eample: The inverse is called a logarithm function log 6 Two raised to what power is 6? The most commonly used bases for logs are 0: log0 log and e: log e ln y ln is called the natural log function. y log is called the common log function.

7 Properties of Logarithms log a a log a a a 0, a, 0 Since logs and eponentiation are inverse functions, they un-do each other. Product rule: log y log log y a a a Quotient rule: log log log y a a a y Power rule: log y log Change of base formula: a log a a ln ln a y

8 Trigonometric Functions Photo by Vickie Kelly, 008 Black Canyon of the Gunnison National Park, Colorado Greg Kelly, Hanford High School, Richland, Washington

9 Even and Odd Trig Functions: Even functions behave like polynomials with even eponents, in that when you change the sign of, the y value doesn t change. cos cos Cosine is an even function because: Secant is also an even function, because it is the reciprocal of cosine. Even functions are symmetric about the y - ais.

10 Even and Odd Trig Functions: Odd functions behave like polynomials with odd eponents, in that when you change the sign of, the sign of the y value also changes. sin sin Sine is an odd function because: Cosecant, tangent and cotangent are also odd, because their formulas contain the sine function. Odd functions have origin symmetry.

11 Shifting, stretching, shrinking the graph of a function Vertical stretch or shrink; reflection about -ais a is a stretch. Vertical shift Positive d moves up. y a f b c d Horizontal stretch or shrink; reflection about y-ais b is a shrink. Horizontal shift Positive c moves left. The horizontal changes happen in the opposite direction to what you might epect.

12 Amplitude and period in trigonometric functions A is the amplitude. Vertical shift B f Asin C D B is the period. Horizontal shift 4 B A C D y sin 4

13 Invertibility of trigonometric functions Trig functions are not one-to-one. However, the domain can be restricted for trig functions to make them one-to-one. y sin These restricted trig functions have inverses.

14 Continuity Grand Canyon, Arizona Photo by Vickie Kelly, 00 Greg Kelly, Hanford High School, Richland, Washington

15 Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your pencil. A function is continuous at a point if the limit is the same as the value of the function. 4 This function has discontinuities at = and =. It is continuous at =0 and =4, because the one-sided limits match the value of the function

16 Removable Discontinuities: (You can fill the hole.) Essential Discontinuities: jump infinite oscillating

17 Removing a discontinuity: f lim has a discontinuity at. Write an etended function that is continuous at. lim f,, Note: There is another discontinuity at not be removed. that can

18 Removing a discontinuity: f,, Note: There is another discontinuity at not be removed. that can

19 Continuous functions can be added, subtracted, multiplied, divided and multiplied by a constant, and the new function remains continuous. Also: Composites of continuous functions are continuous. eamples: y sin y cos

20 Intermediate Value Theorem If a function is continuous between a and b, then it takes f a f b on every value between and. a f b f a b Because the function is continuous, it must take on every y value between and f b. f a

21 Rates of Change and Tangent Lines Devil s Tower, Wyoming Photo by Vickie Kelly, 99 Greg Kelly, Hanford High School, Richland, Washington

22 Slope of a line The slope of a line is given by: 4 f y m The slope at (,) can be approimated by the slope of the secant through (4,6). y We could get a better approimation if we move the point closer to (,). ie: (,9) y Even better would be the point (,4). y 4 y

23 Slope of a line The slope of a line is given by: 4 f y m If we got really close to (,), say (.,.), the approimation would get better still y..... How far can we go? y

24 f h Slope of a line slope f h f y h f h slope at, lim h0 h h f h lim h0 h h h h h lim h0 h y f The slope of the curve at the point is: m lim h0 f a h f a h P a, f a

25 The slope of a curve at a point is the same as the slope of the tangent line at that point. In the previous eample, the tangent line could be found using. y y m If you want the normal line, use the negative reciprocal of the slope. (in this case, ) (The normal line is perpendicular.)

26 Derivatives

27 Derivatives f a h f a h 0 lim h is called the derivative of f at a. We write: f f h f h 0 lim The derivative of f with respect to is h There are many ways to write the derivative of y f

28 f f prime or the derivative of f with respect to y y prime dy d df d the derivative of y with respect to the derivative of f with respect to d f d the derivative of f of

29 4 y f The derivative is the slope of the original function The derivative is defined at the end points of a function on a closed interval y f

30 y y lim h0 y h h h h h 0 lim h 0 y lim h h0 y

31 Differentiability A function is differentiable if it has a derivative everywhere in its domain. It must be continuous and smooth. Functions on closed intervals must have one-sided derivatives defined at the end points.

32 Differentiability To be differentiable, a function must be continuous and smooth. Derivatives will fail to eist at: f f corner cusp f vertical tangent discontinuity f, 0, 0

33 Rules for Differentiation Colorado National Monument Photo by Vickie Kelly, 00 Greg Kelly, Hanford High School, Richland, Washington

34 If the derivative of a function is its slope, then for a constant function, the derivative must be zero. d d c 0 eample: y y 0 The derivative of a constant is zero.

35 d d Derivatives of monomials d d lim h0 d 4 d lim lim h0 h h h h lim h0 h0 h h h lim h0 h h h h 6 h 4h h h h (Pascal s Triangle) 4 We observe a pattern:

36 Derivatives of monomials We observe a pattern: d n d n n eamples: 4 f y 8 f 4 y 8 7 power rule

37 d d Constant multiple rule: cu du c d eamples: d c d d d n cn n When we used the difference quotient, we observed that since the limit had no effect on a constant coefficient, that the constant could be factored to the outside.

38 Sum and difference rules: d u du dv d d d v u v d du dv d d d 4 y y 4 4 y dy d (Each term is treated separately) 4 4

39 Product rule: d uv u dv v du d d d Notice that this is not just the product of two derivatives. d d d d d d

40 Quotient rule: du dv v u d u d d d v v d 5 d 6 5 5

41 Derivatives of trigonometric functions d d sin cos d d cot csc d d cos sin d d sec sec tan d d tan sec d d csc csc cot

42 Derivatives of Eponential and Logarithmic Functions d d e e d d a a ln a d ln d d d log a ln a Mt. Rushmore, South Dakota Photo by Vickie Kelly, 007 Greg Kelly, Hanford High School, Richland, Washington

43 Chain rule dy d dy du du d f g If is the composite of and, then: y f g f g at at ug f u u g

44 Eample y sin 4 d y cos 4 4 d y cos 4 Differentiate the outside function... then the inside function At, y 4

45 Eample d d cos d cos d d cos cos d d cos sin d cos sin 6cos sin It looks like we need to use the chain rule again! The chain rule can be used more than once. (That s what makes the chain in the chain rule!)

46 Higher Order Derivatives: dy y is the first derivative of y with respect to. d y d d d d dy d dy d y is the second derivative. (y double prime) y dy d is the third derivative. y 4 d d y is the fourth derivative.

47 Etreme Values of Functions

48 Global and Local etrema Absolute etreme values are either maimum or minimum points on a curve. They are sometimes called global etremes. They are also sometimes called absolute etrema. (Etrema is the plural of the Latin etremum.) A local maimum is the maimum value within some open interval. A local minimum is the minimum value within some open interval.

49 Global and Local etrema Absolute maimum (also local maimum) Local maimum Local minimum Notice that local etremes in the interior of the function f occur where is zero or is undefined. f

50 Local Etreme Values: If a function f has a local maimum value or a local minimum value at an interior point c of its domain, and if f eists at c, then f c 0

51 4 4 y y y dy d 4 4 -

52 4 4 y dy d First derivative (slope) is zero at: 0,,

53 Critical Point: A point in the domain of a function f at which or f does not eist is a critical point of f. f 0 Note: Maimum and minimum points in the interior of a differentiable function always occur at critical points, BUT critical points are not always maimum or minimum values.

54 Critical points are not always etremes! y f 0 (not an etreme) -

55 y / f is undefined. (not an etreme) -

56 Finding absolute etrema Find the absolute maimum and minimum values of f on the interval,. / / f f f There are no values of that will make the first derivative equal to zero. The first derivative is undefined at =0, so (0,0) is a critical point. Because the function is defined over a closed interval, we also must check the endpoints.

57 At: 0 Finding absolute etrema / f f 0 0 f f At: At: D, f.5874 f To determine if this critical point is actually a maimum or minimum, we try points on either side, without passing other critical points. Since 0<, this must be at least a local minimum, and possibly a global minimum. Absolute minimum: Absolute maimum: 0,0,.08

58 Finding Maima and Minima Analytically: Find the derivative of the function, and determine where the derivative is zero or undefined. These are the critical points. Find the value of the function at each critical point. Find values or slopes for points between the critical points to determine if the critical points are maimums or minimums. 4 For closed intervals, check the end points as well.

59 Using Derivatives for Curve Sketching

60 Rules First derivative: y is positive Curve is rising. y is negative Curve is falling. y is zero Possible local maimum or minimum. Second derivative: y is positive Curve is concave up. y is negative Curve is concave down. y is zero Possible inflection point (where concavity changes).

61 Eample: Graph y 4 There are roots at and. y 6 Set y 0 Possible etreme at 0,. We can use a chart to organize our thoughts. First derivative test: , y y 6 negative y 6 9 positive y 6 9 positive

62 Eample: Graph y 4 There are roots at and. y 6 Possible etreme at 0,. Set y 0 First derivative test: , y maimum at 0 minimum at

63 Eample: Graph y 4 There are roots at and. Possible etreme at 0,. y 6 Or you could use the second derivative test: y 66 y Because the second derivative at = 0 is negative, the graph is concave down and therefore (0,4) is a local maimum. y Because the second derivative at = is positive, the graph is concave up and therefore (,0) is a local minimum.

64 Eample: Graph y 4 We then look for inflection points by setting the second derivative equal to zero. y Possible inflection point at. y 0 y negative y positive There is an inflection point at = because the second derivative changes from negative inflection to positive. point at

65 y y y Make a summary table: rising, concave down local ma 0 falling, inflection point local min 4 9 rising, concave up

66 Definite Integrals Greg Kelly, Hanford High School, Richland, Washington

67 Riemann sum V t 8 When we find the area under a curve by adding rectangles, the answer is called a Riemann sum. 0 4 subinterval The width of a rectangle is called a subinterval. The entire interval is called the partition. partition Subintervals do not all have to be the same size.

68 Riemann sum V t 8 If the partition is denoted by P, then the length of the longest subinterval is called the norm of P and is denoted by P. 0 4 subinterval partition P As gets smaller, the approimation for the area gets better. Area lim P 0 n k f c k k if P is a partition of the interval ab,

69 Definite integrals f c k 0 lim P k n k is called the definite integral of f ab, over. If we use subintervals of equal length, then the length of a subinterval is: b a n The definite integral is then given by: n lim f c k f d n a k b

70 Definite integrals upper limit of integration Integration Symbol b a f d lower limit of integration integrand variable of integration (dummy variable) It is called a dummy variable because the answer does not depend on the variable chosen.

71 Area lim P 0 n k f c k k b a f d Definite integral F( b) F( a) Where F is a function : df d f () F is called indefinite integral

72 The Fundamental Theorem of Calculus If f is continuous on ab,, then the function F a f t dt has a derivative at every point in ab,, and df d d d a f t dt f

73 4 Eample y F A Find the area under the curve from = to =. d c The indefinite integral is defined up to a constant c 0 Proof: df d 0 f ( ) A d F() F() 8 7

74 Eample Find the area between the -ais and the curve 0 y from to. cos d 0 cos cos d sin sin sin 0 sin sin 0 - pos. = = / / sin 0 / neg. = = F This because df sin c Proof: cos f ( ) d

75 Rules for integrals.. a b a 0 a f d b a f d f d Reversing the limits changes the sign. If the upper and lower limits are equal, then the integral is zero.. b a b a k f d k f d Constant multiples can be moved outside. 4. b b b f g d f d g d a a a b c c 5. f d f d f d a b a Integrals can be added and subtracted. Intervals can be added (or subtracted.)

76 Integration by Substitution The chain rule allows us to differentiate a wide variety of functions, but we are able to find antiderivatives for only a limited range of functions. We can sometimes use substitution to rewrite functions in a form that we can integrate.

77 Eample : 5 d Let u u 5 du 6 6 u C 6 6 C du d The variable of integration must match the variable in the epression. Don t forget to substitute the value for u back into the problem!

78 Eample d Let u u du u C The derivative of. du d is d C

79 Eample : 4 d Let u4 u du 4 u C 4 6 u C du 4 d 4 du d 4 6 C

80 Eample 4 cos 7 5 cos u du 7 d Let u75 du 7 d 7 du d sin 7 u C sin C

81 Eample 5 sin d sin cos u du u C du Let u d du d d We solve for because we can find it in the integrand. cos C

82 Eample 6 4 sin cos d sin 4 cos d Let u sin u 4 du du cos d 5 5 u C sin 5 5 C

83 Integration By Parts Start with the product rule: d uv u dv v du d d d d uv u dv v du d uv v du u dv u dv d uv v du u dv d uv v du u dv uv v du u dv d uv v du This is the Integration by Parts formula.

84 Integration By Parts u( ) dv( ) d d u dv uv v du u( ) v( ) du( ) d v( ) d Start with the product rule: d d u( ) u( ) v( ) dv( ) d dv( ) d d d du( ) d u( ) v( ) d d v( ) u( ) v( ) d u( ) d dv( ) u( ) d du( ) v( ) d du( ) d v( ) d

85 u( ) dv( ) d d u( ) v( ) du( ) d v( ) d u can be always differentiated v is easy to integrate. The Integration by Parts formula is a product rule for integration.

86 Eample cos d Easy to integrate u sin v d v d du v u d d dv u ) ( ) ( ) ( ) ( ) ( ) ( d dv cos d du C d d cos sin sin sin cos

87 Eample ln d ln u v d dv d du d v d du v u d d dv u ) ( ) ( ) ( ) ( ) ( ) ( C d d ln ln ln

88 Eample e d Easy to integrate u du d dv d v e e e e e e d Easy to integrate e d u* dv * e d du * v* e d e e e C

89 Taylor Series

90 Suppose we wanted to find a fourth degree polynomial of the form: P a a a a a that approimates the behavior of f ln at 0 If we make P 0 f 0, and the first, second, third and fourth derivatives the same, then we would have a pretty good approimation.

91 P a a a a a 4 f ln 0 4 ln f f 0 ln 0 f f 0 f f 0 P a a a a a P a a0 0 P a a a 4a 4 P 0 a a P a 6a a 4 P 0 a a

92 P a a a a a 4 f ln 0 4 f f 0 P 6a 4a 4 P 0 6a a 6 f 4 f P 4a P a4 4 a 6 4

93 P a a a a a 4 f ln P P 4 0 f ln If we plot both functions, we see that near zero the functions match very well! f P

94 Maclaurin Series: (generated by f at 0 ) f 0 f 0 P f 0 f 0!! If we want to center the series (and it s graph) at some point other than zero, we get the Taylor Series: Taylor Series: (generated by f at ) a f a f a P f a f a a a a!!