MATH 1325 Business Calculus Guided Notes

Transcription

1 MATH 135 Business Calculus Guided Notes LSC North Harris By Isabella Fisher

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3 Section.1 Functions and Theirs Graphs A is a rule that assigns to each element in one and only one element in. Set A Set B Set A Set B The of a function is the set of all. The of a function is the set of all. We can find the value of an element/input in a function by substituting the input into the function and simplifying (this is also known as finding the function value). E1: Given g( ) Find the following: a: g (0) b: g( 1) c: ga ( ) d: g( a) e: g ( 1)

4 t E: Given st () Find the following: t 1 a: s (4) b: s (0) c: sa ( ) d: s( a) e: st ( 1) Sometimes we have functions that are defined by different rules over parts of the domain. They are called. if 1 0 E3: Given f( ) Find the following: if 0 a: f ( ) b. f () c: f (0) E4: Find the domain, range and function values given a graph: D f = Recall to find domain given the graph of a function you: R f = Recall to find range given the graph of a function you: Find f (0) = Find value for for which: f( ) 5 -- = Find value for for which: f( ) 0 -- =

5 If we are not given the graph a function we can determine the domain of a function as follows: 1. Begin by assuming that the domain is ALL REAL NUMBERS ( ). Determine if there is any place where this function is not defined (gives you issues). These issues will be ecluded from your domain. The two types of functions that have issues are: a. Rational functions because division by zero is undefined. b. EVEN root functions because we cannot take the even root of a negative number. A. To find the domain of rational functions: 1. Set the denominator equal to zero and solve for.. The domain will be all real numbers EXCEPT where the denominator equals zero. B. To find the domain of EVEN root functions: i. If the radical is in the numerator: Set the epression under the radical greater than or equal to zero and solve for. ii. If the radical is in the denominator: Set the epression under the radical greater than zero and solve for. (can t be equal to zero because it is in the denominator). Find the domain of the following functions.. 3 EX5: f( ) 5 EX6: f( ) 3 1 EX7: f ( ) 5 EX8: f( ) 1 5 f ( ) 1 EX9: 5 EX10: 1 f( ) ( )( 3)

6 Graph the following functions and find the domain and range of each: EX11: f ( ) 1 EX1: f ( ) EX13: f ( ) 4 EX14: if 0 f( ) 1 if 0

7 Section. The Algebra of Functions Composition of functions: Given functions f() and g() we can determine: i. (f g)() = f(g()) which means to replace all the s in f() with the epression g() is equal to. ii. (g f)() = g(f()) which means to replace all the s in g() with the epression f() is equal to. Use the given functions to find (f g)() and (g f)() E1: f ( ) 1 and g( ) E: f ( ) 3 and g ( ) 1 E3: f ( ) 1 and g ( ) 1 1 Evaluate h () where h g f E4: f 3 ( ) 1 and g 3 ( ) 3 1 E5: 1 f( ) 1 and g ( ) 1

8 Find f ( a h) f ( a) for the following functions 1 E6: f ( ) 3 E7: f ( ) 1 Find E8: f ( a h) f ( a) for the following functions ( where h 0) h f ( ) 1 E9: f ( ) 1 The Cost Function is the variable cost plus the fied cost. The Function is the revenue function minus the total cost function. (If this is negative it represents a loss instead of a profit for a company.)

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11 Section.3 Functions and Mathematical Models Objective: Look at the Cost, Revenue and Profit functions and evaluate function models Analyze the demand and supply curve and determine equilibrium quantity and price

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17 Section.4 Limits We say that a function, f, has a limit L as approaches a lim f ( ) L if the value of f( ) can be made as a close to the number L, taking close to but not equal to a So as gets close to,from both sides, is getting close to a value,. NOTE: if you are NOT getting close to one y value, we say that the limit does not eist DNE Use the graph to determine the limit: EX1: lim f( ) 1 EX: lim f( ) 3 lim f( ) EX3: lim f( ) Find a limit from a table of values 1 EX4: f( ) ; lim f( ) 1 1 f()

18 EX4: f( ) 1 ; lim f( ) 1 f() To find the limit of a function as approaches a finite value, the given value into the function and simplify. Find the indicated limit EX5: lim 3 EX6: lim 3 EX7: lim1 1 EX8: 1 lim EX9: 8 lim 1 4 Sometimes the number you substitute into the function yields 0, this is called form 0 To find the limit of the function where the value causes you to be in indeterminate form you: 1. Factor and Reduce the function. Then take the limit Find the indicated limit EX10: 1 lim 1 1 EX11: lim 0

19 Limit of a function at ± infinity The function, f, has a limit L, as increases without bound lim f ( ) L if f( ) is getting closer to L as approaches infinity. Similarly: lim f ( ) M if f( ) is getting closer to M as approaches negative infinity. Find the indicated limit: EX1: lim f( ) EX13: lim f( ) lim f( ) lim f( ) Theorem (property for a limit at infinity) For all n 0 1 lim 0 n and 1 lim 0 n provided that 1 n is defined. To evaluate the limit for a rational function at ± infinity, divide each term in the numerator and denominator by the highest power of and use the theorem above. Find the indicated limit 1 EX14: lim EX15: lim

20 5 3 1 EX16: lim 6 1 EX17: lim EX18: 4 lim 1 EX19:

21 One Sided Limits: Section.5 One Sided Limits and Continuity A function, f, has the right hand limit, L, as approaches a from the right lim Similarly: a f can be made as close to L with values sufficiently close to a (right of a ) A function, f, has the left hand limit, M, as approaches a from the left lim a f can be made as close to M with values sufficiently close to a (left of a ) f L if the values of f M if the values of The lim a if and only if lim f L AND lim f L a a f L Find the limit EX1: lim lim lim f f f EX: lim 1 lim 1 lim 1 f f f EX3: lim 1 lim 1 f f lim 1 f

22 EX4: lim 3 lim 3 lim 3 f f f Again we see that the lim a if and only if lim f L AND lim f L a a f L To find the limits given a function we: 1. Substitute the given into the function to find the value. If the value causes you to be in indeterminate form, factor and reduce the function, then substitute.. Find the limit Find the indicated limit EX5: lim EX6: lim 0 1 EX7: 1 lim 1 1 EX8: 4 lim EX9: Given f( ) 3 4 if if 0 find the following lim 0 f( ) = lim 0 f( ) = lim 0 f( ) =

23 A function, f, is continuous at a point, a, if it has no holes, jumps, breaks or gaps at a You will usually encounter holes/jumps/breaks with To show that a function is continuous at a point ALL the following points MUST be shown: 1. f( a ) is defined: lim f f a. The f( ) eists: 3. The lim ( ) ( a) a Determine the values of, if any, where the function is discontinuous. State the conditions that are violated if discontinuous. EX10: if 1 f( ) 4 if 1 EX11: 5 if 0 f ( ) if 0 5 if 0 EX1: f 3 ( ) 1

24 Find the values of for which each function is continuous. EX13: f( ) 1 EX14: f( ) 1 EX15: if 1 f( ) 1 if 1 Find the values of for which each function is continuous. 3 EX16: f( ) 4 EX17: f( ) 3

25 Section.6 The Derivative A line to a curve is a line that touches a curve, or a graph of a function, at only a single point. What is the slope of the tangent line T 1? What is the slope of the tangent line T? The derivative geometrically represents the slope of the tangent line at a point to a given function. The derivative also represents the instantaneous rate of change of a function at a point. The problem of finding the rate of change of 1 quantity with respect to another, is the same as finding the slope of the tangent line to a curve Given the following curve: Find the slope of the line between the points P and Q This the called the difference quotient. We say that the slope of a tangent line to a graph at a point P, f is given by h 0 f lim h f f h f The difference quotient measures the the average rate of change of y with respect to over an h interval, h. Since we want to find the average rate of change at a single point, our h gets closer and closer to zero (hence the limit) and this is referred to as the rate of change of f at. h The derivate of a function, f, with respect to is called (read ) The domain of f is the set of all for which the limit esists.

26 Find f E1: f ( ) 7 E: f ( ) 3 4 Find the slope of the tangent line to the graph of the function at the given point. Also determine the equation of the tangent line. E3: f ( ) 3 at, E4: f( ) at 3, 3

27 E5: Let f ( ) 1 Find the following: (a) Find the derivative, f of f (b) Find the equation of the tangent line to the curve at the point (1,3) (c) Sketch the graph E6: Let f ( ) Find the following (a) Find the average rate of change of y with respect to in the interval: = to =3 = to =.5 = to =.1

28 (b) Find the instantaneous rate of change of y with respect to at

29 Section 3.1 Basic Rules of Differentiation Recall that: f h f lim h 0 h f We will learn shortcuts to finding the derivative of a function. Note: d f said d, d of f of d represents the derivative of the function, f, with respect to. d d Rule 1 - The Derivative of a Constant: c 0 The derivative of a constant is ZERO Find d d EX1: f 3 EX: f d n d n n1 Rule The Power Rule: If n is any real number then, d Find d f 5.1 EX3: EX4: f EX5: f 7 4

30 d d d d Rule 3 The Derivative of a Constant multiple of a function: c f c f d Find d f 3 1 EX6: EX7: f 7 EX8: EX9: f 3 EX10: f f r r d d d d d d Rule 4 The Sum Rule: f g f g d Find d EX11: f 5 3 EX1: f EX13: f EX14: 5 4 f 4 3 EX15: f t t t EX16: f t 4 3

31 f 4 find: EX17: let 3 (a) f (b) f 0 (c) f EX18: Find the slope and an equation of a tangent line to the graph of the function, f, at the specified point. f 3 4 ;,6 Find the point on the graph of f where the tangent line is horizontal. 3 EX19: f 3 EX0: f 4 EX1: The demand function for walkie-talkies is given by: demanded in thousands and p is the unit price in dollars. (a) Find f p f where is quantity (b) What s the rate of change of the unit price when quantity demanded is 10,000 units? (c) What s the unit price at that demand?

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33 Section 3. The Product and Quotient Rules d d (the first times the derivative of the second plus the second times the derivative of the first) The Product Rule: given f g f g f g g f Find the derivative: EX1: f 1 EX: f 3 1 EX3: f w w 3 w w1w EX4: f EX5: f 5 1 1

34 f d f g f f g The Quotient Rule: given g d g g (bottom times the derivative of the top minus top times the derivative of the bottom all over bottom squared) Find the derivative: 1 EX6: f EX7: f s s 4 s 1 Find the derivative and evaluate f 1 EX8: 4 f at the given point: ; 1 Find the slope and the equation of the tangent line to the graph at the given point: 4 EX9: f ;, 1 3

35 Find the points on the graph where the tangent line is horizontal: EX10: f 1 Find the points on the graph where the tangent line is equal to 1 : EX11: f EX1: The demand function of eercise equipment is given by d 0 0 where (measured in units of a thousand) is the quantity demanded per week, and d ( ) is unit price in dollars. Find d ( )

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37 The Chain Rule: Section 3.3 The Chain Rule d given g f then g f g f f d d f then f n f f d n The General Power Rule: Find f n n1 4 EX1: f EX: 4 f 1 f EX4: f 7 EX3: 5 EX5: f 7 3 EX6: f 1 3

38 4 EX7: f t t 3 4t 4 EX8: f 1 1 EX9: f 3 3 EX10: f 1 3

39 f EX11: 1 1 Find the equation of the line tangent at the given point EX1: f 7 at pt 3,15

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41 Section 3.4 Marginal Functions in Economics The actual cost incurred in producing an additional unit of a certain commodity given that a plant is already at a certain level of operation is called the marginal cost. The marginal cost is approimated by the rate of change of the total cost function evaluated at the appropriate point. (The marginal cost function is the derivative of the total cost function).

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43 Section 3.5 Higher Order Derivatives The nd derivative of a function, f, is the derivative of the first derivative. By definition, it is: lim h0 f ( h) f ( ) h (tells us acceleration) In general the nth derivative is found by taking the derivative of the (n-1)st derivative. To find f take the derivative of f To find f take the derivative of f To find f '''' take the derivative of f Find the 1 st and nd derivative of the following functions. 3 E1: f ( ) 3 1 E: f ( ) 1 E4: h( ) 1 1 g( t) t 3t 1 E3: 4

44 h( w) w w 4 E6: E5: 5 t gt () t 1 Find the 3 rd derivative of the following function E7: f ( ) 1

45 Solve the following

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47 Section 3.6 Implicit Differentiation and Related Rates Currently we only know how to differentiate if we have a function all in the same variable, i.e, f ( ) 16 Implicit differentiation allows us to take the derivative of an equation that has more than one variable, i.e, y 16. We call it implicit differentiation because the equation is not eplicitly solved for y. d d derivative of d d derivative of with respect to Steps for implicit differentiation: 1. Differentiate both sides of the equation with respect to. (make sure that the derivative of any term involving y includes the factor dy d. Find dy by implicit differentiation d E1: y 16. Solve the resulting equation for dy d and y. in terms of Find dy by implicit differentiation d 3 3 E: y y 4 0 E3: y 6

48 E4: y y 8 E5: 1 1 y 1 3y E6: 1 3 Find the slope and an equation of a tangent line to the graph of the given equation at the specified point E7: 4 9y 36 ; 0,

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51 Section 4.1 Applications of the First Derivative A function, f, is increasing on an interval ab, if for every two numbers 1 and whenever 1 (increasing if from left to right, the graph is going up) A function, f, is decreasing on an interval ab, if for every two numbers 1 and in ab,, f f in ab,, f 1 f, whenever 1 (decreasing if from left to right, the graph is going down) in, A function, f, is constant on an interval ab, if for every two numbers 1 and (constant if from left to right, the graph does NOT go up or down, it remains the same) 1 ab, f f, 1 We can use f to determine intervals of increasing/decreasing/constant: If f 0 for every value of in the interval, If f 0 for every value of in the interval, If f 0 for every value of in the interval, ab then f is increasing on ab, ab then f is decreasing on ab, ab then f is constant on ab, To determine the intervals of increasing/decreasing/constant: 1. Find all the values of for which f 0 and where f is discontinuous (undefined) and identify the open intervals determined by the numbers (these are called critical numbers). Select a test value, c, in each interval found in step 1 and determine the sign (+/-) and determine the sign of c f in that interval: the function f is increasing a. If f c 0 the function f is decreasing b. If f c 0

52 Find the interval(s) where the function is increasing and the interval(s) where it is decreasing: 4 3 E1: f ( ) E: f ( ) 4 10 E3: f( ) E4: f ( ) 7 Relative Etrema give us the highest points and lowest points in an interval. Relative Maimum: A function, f, has a relative ma at c if there eists an open interval ab, containing c such that f f c for all in ab, Relative Minimum: A function, f, has a relative min at c if there eists an open interval ab, containing c such that f f c for all in ab, To find the relative ma/min of a function: 1. Determine the critical numbers (any number in the domain of f, such that f 0 or. Determine the sign (+/-) of f a. If f b. If f c. If f to the left and right of each critical number changes sign from positive to negative (+ to ) then c is a relative ma changes sign from negative to positive ( to +) then c is a relative min DOESN T change sign then f doesn t have relative etrema at c f DNE )

53 Find the -value(s) of the relative maima and relative minima, if any, of the function. 4 3 E5: f ( ) 6 6 E6: f ( ) 4 8 E7: 9 f ( ) f ( ) 1 E8: 3

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55 Section 4. Applications of the Second Derivative The nd derivative determines concavity. Concavity of a Function f : Let the function f be differentiable on an interval ab,. Then, 1. The graph of f is concave upward on,. The graph of f is concave downward on, Theorem : If f 0 for every value of in, If f 0 for every value of in, ab if f is increasing on ab,. ab if f is decreasing on ab,. ab then the graph of f is concave upward on ab, ab then the graph of f is concave downward on ab, To determine intervals of concavity of the graph of f 1. Determine the values of for which f is zero or is not defined, and identify open intervals determined by these numbers.. Determine the sign (+/-) of f in each interval found in step 1 compute f then the graph of f is concave upward a. If f 0 then the graph of f is concave downward b. If f 0 Determine where the function is concave upward and where it is concave downward. E1: f ( ) 3 4 E: f 4 3 ( ) 6 8

56 E3: f ( ) 3 An inflection point is a point on the graph of a continuous function f, where the tangent line eists and the concavity changes. To determine inflection points: 1. Compute f. Determine the numbers in the domain of f for which f 0 or where f 3. Determine the sign of f the sign (+/-) as we move across computing f c does not eist. to the left and right of each number, c (found in step ). If there is a change in c, then c will be an inflection point c, f c, which is found by Find the inflection point(s), if any, of the function 1 E4: f ( ) E5: g 4 3 ( ) 3 4 1

57 Instead of using the first derivative to determine relative maimums/minimums we can use the second derivative test. Second Derivative Test: 1. Compute f and f ). Find all the critical numbers of f at which f 0 3. Compute f c for each critical number c (the numbers we found in step ) then f has a relative maimum at c a. If f c 0 then f has a relative minimum at c b. If f c 0 c. If f c 0 OR f c DNE, then the test has failed and is inconclusive Use the Second Derivative Test to find the relative etrema, if any, of the function E6: f ( ) 5 10 E7: f ( ) 3

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59 Section 4.3 Curve Fitting We will be learning how to sketch the graph of a function by using application of the 1 st and nd derivatives. Vertical Asymptotes: A function, f, has a vertical asymptote at the line a, if: lim a or lim f or a f or What we will do to find vertical asymptotes is: 1. Factor and reduce the function if possible. Set the denominator equal to zero and solve for. Any REAL NUMBERS you obtained is where the function has a vertical asymptote. f 1 Horizontal Asymptotes: A function, f, has a horizontal asymptote at the line y b, if: or lim lim f b To find the horizontal asymptote: 1. Take the limit as the function approaches infinity. f 1 f b Find the vertical and horizontal asymptotes of the following functions:

60 1 EX1: y EX: y 3 1 EX3: g t t t 9 EX4: f 5 f Note: The graph of a polynomial function does NOT have asymptotes EX5: To sketch the curve of a function: 1. Find the domain of the function. Find the and y intercepts of the function 3. Determine the end behavior of (behavior of f for large absolute values of ) 4. Find all the horizontal and vertical asymptotes of the function 5. Determine intervals of increasing and decreasing 6. Find the relative etrema 7. Determine concavity 8. Find the inflection points 9. Plot all the information above and any additional points necessary and sketch the graph. Sketch the graph of the following functions:

61 f 3 1 EX6: 3 a) Domain: b) X-intercept(s): y-intercept: c) End Behavior: d) Horizontal Asymptote: Vertical Asymptote: e) Increasing: decreasing: f) Relative Etrema: g) Concave up: Concave down: h) Inflection points:

62 EX7: g 1 a) Domain: b) X-intercept(s): y-intercept: c) End Behavior: d) Horizontal Asymptote: Vertical Asymptote: e) Increasing: decreasing: f) Relative Etrema: g) Concave up: Concave down: h) Inflection points:

63 EX8: f 1 1 a) Domain: b) X-intercept(s): y-intercept: c) End Behavior: d) Horizontal Asymptote: Vertical Asymptote: e) Increasing: decreasing: f) Relative Etrema: g) Concave up: Concave down: h) Inflection points:

64 EX9: f t t t 9 a) Domain: b) X-intercept(s): y-intercept: c) End Behavior: d) Horizontal Asymptote: Vertical Asymptote: e) Increasing: decreasing: f) Relative Etrema: g) Concave up: Concave down: h) Inflection points:

65 Section 4.4 Optimization I Absolute Etrema of a Function o If f f c for all in the domain of f then o If f f c for all in the domain of f then f c is called the absolute maimum value of f f c is called the absolute minimum value of f The Etreme Value Theorem o If a function f is continuous on a closed interval ab, then f has both an absolute maimum value and an absolute minimum value on ab, (this is because on a closed interval there will be a highest and lowest point on the graph) We find absolute etrema on an open interval eactly as we did with relative etrema: 1. Find the first derivative. Find the critical numbers 3. Determine if there is a sign change (+/-) between the critical numbers

66 To find the absolute etrema of f on a closed interval ab, then: 1. Find the first derivative. Find the critical numbers ( ) 3. Compute f a and f b and f critical # s 4. The absolute maimum value and the absolute minimum value will correspond to the largest and smallest numbers, respectively, found in step 3. Find the absolute maimum value and the absolute minimum value, if any, of the function E1: f E: g 4 6 on 0,5 E3: E4: h f 1

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69 Section 4.5 Optimization II Guidelines to solving optimization problems: 1. Assign a letter to each variable mentioned in the problem and draw a picture if necessary.. Find an epression for the quantity that will be optimized. 3. Use the conditions in the problem to find the quantity to be optimized as a function of 1 variable. 4. Optimize the function over its domain.

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71 Section 5. Logarithmic Functions A logarithm of to the Base b is defined by: y log b if and only if y b b 0, b 1, and 0 For Eample: 3 8 Common logarithms: log is the same as ln is the same as thus log thus log log10 - if no base is written the implied base is 10 log e - ln means log base e Laws of Logarithms: Given m and n are positive real numbers and b 0 and b 1 then: 1. log b mn = log m log n m. log b n b = log m log n n 3. logb m = n log b m 4. logb log b 1 b Use the laws of logarithms to epand and simplify the epression. EX1: log 1 4 b b b 1 EX: log 1 EX3: ln e EX4: ln 1 1

72 The derivative of an Eponential Function is: Section 5.4 Differentiation of Eponential Functions d e e d The chain rule for Eponential Functions is: f d e e f d Find the derivative of the following functions. EX1: f 3e EX: f e EX3: f e u EX4: f u u e e EX5: f EX6: f e e 1

73 EX8: f s s 1e s f 4 e 3 EX7: 3 Find the nd derivative of the following function EX9: 3 t t f t e 5e Find the intervals where the following function is increasing and where it is decreasing EX10: f e

74 Determine the intervals where the function is concave down and/or concave up. EX11: f e

75 Section 5.5 Differentiation of Logarithmic Functions The derivative of d d ln is: ln 0 1 The chain rule for ln is: d ln f f f 0 d f The derivative of the function you re taking the natural log of, divide by the function. Find the derivative of the following functions. f ln 1 EX1: 8 EX: f ln f 1 ln EX3: EX4: f ln EX5: f ln 1 EX6: f ln

76 EX7: f ln EX8: f ln 3 EX9: f ln EX10: f e ln Find the second derivative of the following function: EX11: f ln

77 We can find the derivative of functions by taking the natural log of both sides of the equation and using the laws of logs to epand before taking the derivative. Finding dy by Logarithmic differentiation: d 1. Take the ln of both sides of the equation and uses the laws of logarithms to write any complicated epressions as a sum/difference of simpler terms.. Differentiate both sides of the equation with respect to 3. Solve for dy d Use logarithmic differentiation to find the derivative of the following functions. EX1: y 1 3 EX13: y EX14: y 3

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79 The eponential growth model is : 0 Section 5.6 Eponential Functions as Mathematical Models Q t kt Q e where 0 t Q 0 is the amount of substance that is initially present when t 0 and k is the constant of proportionality, called the growth constant.

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81 Section 6.1 Antiderivaties and the Rules of Integration -Recall that the derivative allowed us to find velocity given position. Now what we will do is go from knowing the velocity to finding the position. A function, F, is an antiderivative of f on an interval I if F f for all in I For eample let s say that F 1 is an antiderivative of F = f f 4 1 this is because But note that if F it would also be the antiderivative of f 4 1 Also if F we would get the same result. The process of finding antiderivatives is called integration. f d F C Read: the indefinite integral of f of with respect to equals F of plus c The indefinite integral gives a family of functions because of the constant, thus when we take the antiderivative of a function we will be adding a CONSTANT TERM, C to the antiderivative. The function to be integrated is called the integrand and C is called the constant of integration. Rules of integration: 1. The indefinite integral of a constant k d k C (where k is a constant term) E1: d E: d E3: e d n 1 n1. The power rule d C (where n does not equal -1) n 1 3 E4: d E5: d E6: 4 d E7: 3. The indefinite integral of a constant multiple of a function d E8: 5 3 d c f d c f d (where c is a constant) E9: d

82 f g d f d g d (we can integrate a term at a time) 4. The sum rule 3 E10: d 5. The indefinite integral of the eponential function E11: 1 e d e d e C E1: 1 6. The indefinite integral of the function f 5 d 1 d ln C (where does not equal 0 ) Find the indefinite integral of the following functions E13: 3 e d E14: t t 3 3 t dt

83 3 3 E15: 10 u u u du If we are given a point on the graph of f (an initial value), we can find the eact value of C for the antiderivative. Find f by solving the initial value problem. f 3 6 ; f 4 E16: ; 0 E17: f e f E18: Find the function f given that the slope of the tangent line to the graph of f at any point 3 and the graph of f passes through the point 1,., f is

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85 Section 6. Integration by Substitution Integration by substitution allows us to integrate functions that we cannot separate into individual terms. It is related to the chain rule for differentiating. Steps for Integrating by Substitution: 1. Let u equal one of the functions, g, in the integrand (usually the inside function, or the function that is being raised to a power). Find du g d 3. Use substitution to convert the ENTIRE into one only involving u 4. Evaluate the resulting integral using the rules for integration from section Replace u with g to obtain a final solution as a function in terms of Find the indefinite integral 4 1 d E: E1: d E3: d E4: 3 1 e e 3 e e d Sometimes we will need to do a little work to get du to equal E5: e d g d

86 Find the indefinite integral E6: ln u 3 du E7: u 1 ln d E8: e du E9: t dt Hint: t 4 t 4 1 t4 t4

87 Section 6.3 Area and the Definite Integral We are going to use Riemann Sums to approimate the area under the curve of a graph: Let s say we wanted to find the area under f() from a to b. Fig A Fig B The area under the graph of a function can be found as follows: Let f be a nonnegative continuous function on a closed interval ab,. Then the area of the region under the graph of f is A f f f f n where 1,, 3,..., n are arbitrary points in the n subintervals of, ab of equal width b a n The sum on the right-hand side of this epression is called a Riemann sum, in honor of the German mathematician Bernhard Riemann ( ) E1: Find an approimation of the area of the region R under the graph of f by computing the Riemann sum of f corresponding to the partition of the interval into the subintervals shown in the figure. Use the midpoint of the subintervals as the representative points.

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90 Find an approimation of the area of the region R under the graph of the function f on the interval [a,b]. In each case, use n subintervals and choose the representative points as indicated. E4: f n ( ) 4 ; [ 1,] ; 6 ; left endpoints E5: f ( ) e ; [0,3] ; n 5; midpoints

91 Section 6.4 The Fundamental Theorem of Calculus We will use the fundamental theorem of calculus to calculate the area of a region under a graph. The Definite Integral: Let function f be defined on representative points 1,, 3,..., n ab,. If f f f f eists and is the same for all lim n n in n subintervals of, b b a ab of equal width then this area is called the n Definite Integral denoted f d where a the lower limit of integration & b the upper limit of integration a The Fundamental Theorem of Calculus: Let f be a continuous function on a closed interval Where F is any antiderivative of f, that is F f b ab,, then a f d F b F a Find the area of the region under the graph of the function f on the interval ab, E1: f 3 ; 1, 1 f ; 1, E: E3: f 1 3 ; 8, 1

92 Evaluate the definite integral E4: 4 u du E5: d 1 E6: 1 1 d

93 Section 6.5 Evaluating Definite Integrals Properties of the Definite Integral: Let f and g be integrable functions. Then, a 1. f d 0 a b a. a f d f d b b 3. a b cf d c f d where c is a constant a b b b 4. f g d f d g d a a a b 5. c b f d f d f d where a c b a a c Evaluate the definite integral 1 4 d E: 3 E1: d 1 E3: d E4: 0 e d

94 Find the area of the region under the graph of the function f on the interval ab, 1 f ; 1, E5: The Average Value of a Function: Suppose f is integrable on ab,, then the average value of f on ab, is: Find the average value of the function f over interval ab, E6: f 4 ;,3 E7: f e ; 0,4 1 b b a f d a