lim 1 lim 4 Precalculus Notes: Unit 10 Concepts of Calculus
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1 Syllabus Objectives: 1.1 Te student will understand and apply te concept of te limit of a function at given values of te domain. 1. Te student will find te limit of a function at given values of te domain. Limit ow te outputs of a function beave as te inputs approac some value Notation: L lim f f. Te limit (L) as approaces c of c Finding a Limit I. Table sin E: lim 0 Use te table to coose values of close to zero (from te left and rigt). II. As approaces 0, it appears te function is approacing 1. sin lim 1 0 Grapically 4 E: lim Grap te function and use te TRACE feature to see function values as we approac from te left and rigt. III. As approaces, it appears te function is approacing 4. 4 lim 4 Analytically a. Direct Substitution 8 E: lim Substitute into te function: 8 8 lim Page 1 of 1 Precalculus Grapical, Numerical, Algebraic: Pearson Capter 9
2 b. Algebraic Manipulation 9 E: lim 3 3 We cannot use direct substitution, because of division by zero. So we must simplify te function algebraically lim lim lim Noneistent Limits 3 1 E: lim 3 3 We cannot use direct substitution, because of division by zero. Factoring te numerator, we ave: lim lim Tis does not simplify to avoid division by zero So, lim does not eist. 3 3 One-sided Limits 1. Rigt-Hand Limit: te limit of f as approaces c from te rigt. lim f ( ) E: Find lim f, wen f 31 Because we are approacing from te rigt, we must substitute into te piece of te piecewise functions tat represents values of greater tan. lim f lim Note: Te limit value differs from te function value at. c. Left-Hand Limit: te limit of f as approaces c from te left. lim f ( ) E: lim We cannot use direct substitution, because we cannot approac from te left. Te domain of te function f is Two-sided Limit,. So lim does not eist. lim f ( ) c ** f as a limit as approaces c if and only if te rigt and left and limits at c eist and are equal** c Page of 1 Precalculus Grapical, Numerical, Algebraic: Pearson Capter 9
3 E: Use te grap below of f to answer te following questions. a. lim f 1 b. lim f 1 c. lim f 1 d. f 1 1 lim f lim f does not eist lim f f 1 1 If LM,, c, and kare real numbers and lim f( ) L and lim g( ) M, ten c c PROPERTIES OF LIMITS: 1. Sum and Difference Rules: lim( f( ) g( )) LM Te limit of te sum or difference of two functions is te sum or difference of teir limits. c. Product Rule: lim( f( ) g( )) LM c Te limit of a product of two functions is te product of teir limits. 3. Constant Multiple Rule: lim( k f( )) kl c Te limit of a constant times a function is te constant times te limit of te function. f( ) L 4. Quotient Rule: lim, M 0 c g ( ) M Te limit of a quotient of two functions is te quotient of teir limits, provided te limit of te denominator is not zero. 5. Power Rule:If r and s are integers, s 0, ten r r r s s s lim( f( )) L provided tat L is a real number. c Page 3 of 1 Precalculus Grapical, Numerical, Algebraic: Pearson Capter 9
4 tan E: Find lim. 0 Using direct substitution, we divide by zero. So try algebraic manipulation. We can rewrite tan as sin cos. sin tan cos sin sin 1 lim lim lim lim cos 0 cos sin 1 sin 1 1 Using te product rule, lim lim lim cos 0 0cos 1 You Try: Calculate te limit. t lim 3t t 4 QOD: Eplain ow left and rigt-and limits relate to two-sided limits. Page 4 of 1 Precalculus Grapical, Numerical, Algebraic: Pearson Capter 9
5 Syllabus Objectives: 1.3 Te student will understand, interpret and apply te concept of te difference quotient between two points on a curve as te average rate of cange of te function over a given interval of te domain. 1.4 Te student will understand, interpret and apply te concept of te tangent line at a given point on a curve as te instantaneous rate of cange of te function at tat point. Average Velocity: cange in position cange in time or v a s t E: Wat is te average velocity of a racecar tat drives a quarter mile in 4.5 seconds? s 0.5 mi 1 mi 60 sec 60 min 3600 mi Average velocity: va 00 mp t 4.5 sec 18 sec 1 min 1 r 18 r Discuss: Wat was te car s velocity at eactly 4 seconds? More information would be needed to determine te answer. Tis is called instantaneous velocity. Slope: cange in cange in y y and zoom in on te point Discuss: Does a parabola ave a slope? Grap y,4. Yes, a parabola does ave a slope. As we zoom in on te point, te grap appears to be a line. Rate of Cange In linear equations, te rate of cange is constant (te slope) Te slope of te tangent line at a point gives us te rate of cange at tat instant, or te slope of te curve at tat point. Te line sown is a tangent line. Page 5 of 1 Precalculus Grapical, Numerical, Algebraic: Pearson Capter 9
6 Pierre Fermat (169) 1. Start wit te slope of a secant troug P, f and a point Q, f nearby.. Find te limiting value of te secant slope as Q approaces P. 3. Tis is te slope of te curve at P and te slope of te tangent line to te curve at P. Q( +, f ( + )) P(, f ()) Slope of te Secant Line: m sec y f f f f Let P Q, ten 0. Slope of te Tangent Line: m tan f f lim 0 Tangent to a Curve te line troug P wit te slope as calculated above f ( a) f ( a) Slope of te Curve: Te slope of y f at te point Pa, f a is m lim 0 Note: As approaces 0, te two points approac one point. Te slope of te curve at point P is te same as te slope of te tangent line at point P. E: Find te slope of f 10 at 3, f 10 f f f m lim lim lim lim lim lim At m , 1 : Page 6 of 1 Precalculus Grapical, Numerical, Algebraic: Pearson Capter 9
7 Alternate Metod: f ( a) f ( a) Use m lim wit a : f f lim lim lim lim lim f f Derivative: Te derivative of a function f wit respect to : f lim 0 Note: f is a function. (Read f prime of.) Derivative is SLOPE. dy d d d Notation: Derivative of f at = f, y,, D y, f f. E: Find te derivative of 4 4 f lim lim lim lim Calculating a Derivative at a: a f 0 f a lim f a E: Find f 3 if f 4. f 3 f f 3lim lim lim lim Differentiability Wen f a does not eist: a. Corner one-sided derivatives differ E: y at 0 Te left-and derivative at 0 is 1, and te rigt-and derivative at 0 is 1. b. Cusp te slopes of te secant lines approac from one side and from te oter E: y 3 at 0 Te slopes of te secant lines approac from te left of 0, and tey approac from te rigt of 0. c. Vertical Tangent te slopes of te secant lines approac eiter or from bot sides Page 7 of 1 Precalculus Grapical, Numerical, Algebraic: Pearson Capter 9
8 3 E: y at 0 Te slopes of te secant lines approac from bot sides. d. Discontinuity te function as a discontinuity 1 0 E: f at Te function as a jump discontinuity at 0. Teacer Note: Have students grap te functions above to get a visual of eac type. You Try: Find te formula for te slope of,0.5. f 1 4 and ten use it to find te slopes at 0,0.5 and QOD: A differentiable function is always continuous. Is te converse to tis statement true? Eplain. Page 8 of 1 Precalculus Grapical, Numerical, Algebraic: Pearson Capter 9
9 Syllabus Objectives: 1.6 Te student will understand, interpret, and apply te concept of area under a curve as a cumulative total of some quantity. 1.7 Te student will model application problems involving areas under curves. 1.8 Te student will eplore topics connecting iger-level matematics to real-world applications. Recall: Distance = rate time, or d rt, were r is te average rate (speed) s s is used to represent position, so s would be te distance traveled: r t E: If it takes you minutes to travel 1 mile from your ouse, ow fast were you traveling? s 1 mi 60 min r : r 30mp t min 1 r Te Connection to Areas E: Grap te time (in ours) as a function of te velocity. Average Velocity = 30 mp Instantaneous Velocity = 30 mp at t = 1 min v v t t Calculate te areas of eac region Average Velocity: A 30 1 Instantaneous Velocity: A Wat do te areas represent? Eplain. Te areas represent te total distance traveled, 1 mile. To calculate te area, we multiplied velocity (mp) times time (ours). mi r mi r Approimating Areas Under Curves Riemann Sum: te sum of te areas of rectangles tat lie under a curve LRAM: Left Rectangular Approimation Metod RRAM: Rigt Rectangular Approimation Metod Page 9 of 1 Precalculus Grapical, Numerical, Algebraic: Pearson Capter 9
10 f, E: Use Riemann sums to approimate te area of te region bounded by te grap of te -ais, 0, and, using 4 subintervals. Te eigt of eac rectangle is te corresponding function value of te endpoint. Te widt of eac rectangle is te lengt of eac subinterval = 0.5. LRAM Start wit te left-most endpoint. y LRAM: RRAM Start wit te rigt-most endpoint. y A f f f f RRAM: A f f f f Discuss: Wic approimation is better? Describe two ways we could find a better approimation. One approimation is too small (LRAM) and te oter is too big (RRAM). A better approimation could be found by using more rectangles (smaller subintervals). Or we could find te average of te two. Definite Integral: te value of te area under te nonnegative curve f from on te interval ab, Page 10 of 1 Precalculus Grapical, Numerical, Algebraic: Pearson Capter 9
11 Notation: b a f d Read: Te integral of f of d from a to b. b a lim i f d f n n i 1 n = number of subintervals is approacing f i eigt of rectangles at eac interval endpoint widt of eac subinterval If tis limit eists, ten f is integrable on ab,. E: Estimate te area bounded by LRAM: d f f f f f 1 y and te -ais from 1 to, using 5 subintervals RRAM: d f f f f f LRAM: RRAM: Because te LRAM is too big and te RRAM is too small, we can make our estimate better by finding teir average: Numerical Integral: Te calculator can estimate a definite integral using fnint (TI-84). E: Ceck te estimation from te eample above for d. 1 fnint can be found in te MATH menu. Type in fnint(function, variable of integration, lower limit, upper limit) Page 11 of 1 Precalculus Grapical, Numerical, Algebraic: Pearson Capter 9
12 Evaluating a Definite Integral Using Areas 1 E: Evaluate d 4 by computing te area. Grap: Find te area of te saded region in te interval 4,1. y Te region is a trapezoid wit bases of lengt 4 and 6 and a eigt of Atrap b1 b sq units ; So d 4 You Try: Estimate te area bounded by y 4 and te -ais from 0 to using 4 subintervals. Compare your answer wit te estimate from te calculator. QOD: Anoter rectangular approimation metod is called te MRAM (midpoint rectangular approimation metod). Would tis metod give te same result as finding te average of te LRAM and RRAM? Eplain. Page 1 of 1 Precalculus Grapical, Numerical, Algebraic: Pearson Capter 9
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