3.7 Implicit Differentiation -- A Brief Introduction -- Student Notes

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1 Fin these erivatives of these functions: y.7 Implicit Differentiation -- A Brief Introuction -- Stuent Notes tan y sin tan = sin y e = e = Write the inverses of these functions: y tan y sin How woul we fin the erivatives of these inverse functions? y e Let s look at a brief introuction to Implicit Differentiation so that we can fin the erivatives of these three inverse functions. Up to now, we have worke eplicitly, solving an equation for one variable y in terms of another variable. For eample, if you y were aske to fin for y 4, you woul solve for y an get y 4 an then take the erivative. This erivative requires the use of the chain rule. Sometimes it is inconvenient, ifficult or impossible to solve for y. In this case, we use implicit ifferentiation. It is imperative to note that anytime you see a y-variable you must think of y as a function of just as in the notation: know the eplicit form of f() I will apply the chain rule to inicate it s erivative. Differentiating with respect to : y f. Since I o not Variables agree use power rule variables agree y y y Variables isagree use power rule an chain rule variables isagree 5 4 y 0 y variables isagree variables agree

2 Practice:. y Fin 4 y 5. y y cos.6 Implicit Differentiation -- Stuent Notes Let s return to our inverse functions an use implicit ifferentiation to fin their erivatives: arctan arcsin ln Eamples using the erivative rules we just foun an applying rules we alreay learne: (arctan( t )) (arcsin(tan( ))) ln( ) a) b) c) t ) ( t ln t) t e) ( ln( y)) y f) (cos(sin ))

3 Derivative of Inverse Function Theorem Function an Inverse Pre-requisites: Given each function, ientify key points on the function that fall on lattice points of the coorinate gri. Mark 7 points on the parabola with visible ots. Mark points on the cubic with visible ots. ) ) a) Write the equation of each function in (h,k) form an evaluate the function at the given point. Quaratic function, f = Cubic function, f = f f b) For each function, list the operations on that yiel y. Parabola Cubic f c) Write inverse equations by using the list in (b) & applying inverse operations in reverse orer on. State the corresponing inverse coorinate from the point on the function in part (a), f f, f ) Accurately, sketch the inverse function on the coorinate gri using the key lattice points. Label f. f f e) Fin the erivative of the function at the specifie point. Fin the erivative of its inverse at the corresponing point on the inverse. y f ' ' ' ' f f ' y f ' f) What is the relationship between the erivative value of the function at the point an its inverse at the corresponing inverse point?

4 AB Calculus Supplement: Derivative of the Inverse of a Function Suppose that f an g f are inverse functions. What is the relationship between their erivatives? Algebraically: inverses are obtaine by interchanging the an y coorinates an solving for y. Graphically: inverses are reflections of each other in the line y. If f passes through the point ab,, then the slope of the curve at ais represente by f a an by the ratio of the change in y over ' the change in, slope triangle at point y. In the figure, note the P a, b on f with vertical length y an horizontal length an slope y. When the graph of f is reflecte in the line y, we obtain the graph of the inverse of f enote as f an this inverse graph passes through the point ba,. The slope of the inverse curve at bis represente by f ' b an by the ratio of the change in over the change in y, because the y y horizontal an vertical lengths of the slope triangle at Point P were interchange at P. The slope of the line tangent to the graph of g f at b is the reciprocal of the slope of f at a. Given ab, if Derivative of the Inverse of a Function is a point on f an f ' a m g f g ' b f ' b, then is the inverse of f, m. The erivative of the inverse of a function at a point is the reciprocal of the erivative of the function at the corresponing point. 4

5 Eamples: ) If f 7 an f ' 7 5 an g is the inverse of f, that is g f g '?, then ) If f 5, f ' 6 an f ' 5 an g is the inverse of f, that is, then g g f ' 5? ) Values for a function f an its erivative are shown in the table. If g is the inverse of f then evaluate g '4 an ' g. f f ' ) Let f, an let g be the inverse of f. Evaluate ' g. 4 g at 5) Let f, f ' & g be the inverse of f, what is the equation of the tangent line to? 5

6 6) The following figure shows f an f a) f, f, f ', f '.. Using the given table, fin: f f ' b) The equation of the tangent line at the points,8 Q 8,. P an c) What is the relationship between the two tangent lines? 7) Calculate g ' where g is the inverse of f e without solving for g. 8) Calculate g' where g is the inverse of f without solving for g. 4 9) Let f. Assume f is one-to-one, meaning that f function. has an inverse that is also a a) What is the value of f when? at. b) Fin the slope of the tangent line to the curve y f 6

7 Keys to Properly Solving Derivative of Inverse Problems: Ientify the point ab, on the function f using the information that is given. Differentiate f Take the reciprocal of the erivative of f. This is the erivative of Evaluate the erivative of f at the point ba, f. Practice: Given the table of values for ifferentiable functions f an g. a) If h f, then evaluate '4 h. f f ' g ½ ½ g' b) If h f, then evaluate ' h. c) If g, then evaluate '. ) If g, then evaluate '. An these are not erivatives of inverses, but they are goo practice. e) If p g, then evaluate ' p. f) If b f g, then evaluate ' b. g) If n f h) If, then evaluate ' m f p., then evaluate m '9. i) If q g, then evaluate q '. 7

8 .7 Implicit Differentiation -- Stuent Notes Up to now, we have worke eplicitly, solving an equation for one variable in terms of another. For eample, if you y were aske to fin for y 4, you woul solve for y an get y 4 an then take the erivative. Sometimes it is inconvenient or ifficult to solve for y. In this case, we use implicit ifferentiation. You assume y coul be solve in terms of an treat it as a function in terms of. Thus, you must apply the chain rule because you are assuming y is efine in terms of. Differentiating with respect to : Variables agree use power rule variables agree y y y Variables isagree use power rule an chain rule variables isagree y y variables agree variables isagree y y y y variables isagree y y y use prouct an chain rules Consier the problem, fin y for y y. Treat y as a quantity in terms of so y y Different Same y y y y y y Now solve for y Guielines for Implicit Differentiation: y. y y y 8

9 . Differentiate both sies of the equation with respect to.. y Collect all terms involving on one sie of the equation an move all other terms to the other sie. y. Factor out of the terms on the one sie. y 4. Solve for by iviing both sies of the equation by the factore term. Practice: Fin y :. 7cos y y. 4 y y. y y y 0y 0 5. Fin the equation of the normal line (the line perpenicular to the tangent line) to the curve y y 5 at the point (, ). 6 Given y f rewrite the inverse function as erivative of an inverse function. Write the final answer in terms of only. f y then use implicit ifferentiation to fin the 9

10 .9 Linear Approimation an the Derivative Stuent Notes Tangent Line Approimations: We can use the equation of the tangent line to approimate the value of a function at a particular value of. The concavity of the function tells us if an approimation mae with the tangent line if an over-estimate (too high) or an uner-estimate (too low.) If a function is concave up, the tangent line will be tangent line equation will be. If a function is concave own, the tangent line will be tangent line equation will be. the curve an any approimation mae from the the curve an any approimation mae from the Sketch four portions of graphs satisfying the criteria given, then raw a point on each of the portions an raw a tangent line to the curve at that point. Do your pictures illustrate the conclusions you mae above? f ' 0 & f " 0 f ' 0 & f " 0 f ' 0 & f " 0 f ' 0 & f " 0 For each question below, write the equation of the tangent line to the curve at the esignate value of. Use the tangent line equation to approimate the value of the function at the given -value. Finally use the n Derivative an concavity to justify whether the tangent line approimation is too high or too low. Function & a f 49 Tangent line equation at a Tangent line Secon Derivative approimation at evaluate at a a f 50 f " 49 Is the tangent line approimation an overestimate or unerestimate? Justify using f f f. f " 0

11 ln f e f f " e 4 g f. f " h 5 f.0 f " 6 j 6 cos f 0.5 f " 6

12 .0 Mean Value Theorem Stuent Notes MEAN VALUE THEOREM: If a function is continuous on [a, b] an ifferentiable on (a, b), then there is as number c in the interval (a, b) such that f ( c) = f ( b ) f ( a ) b a or (b a) f ( c) = f (b) f (a).. Use the graph to illustrate the Mean Value Theorem with a continuous an ifferentiable function. Show f (), a, b, c an all other conitions of the theorem.. Fin the number c that satisfies the Mean Value Theorem (MVT) for f on the interval [0,4]. Draw a picture.. Why oes the MVT not apply? a) y = on [0,] b) f () = on [-,] 4. Apply the MVT, if possible. If not possible eplain why. A f () = on [-,] B f () = on [0,] C f () = on [0,]

13 MVT Problems. The function f ( ) on [ - 8, 8] oes not satisfy the conitions of the Mean Value Theorem because A. f (0) is not efine B. f () is not continuous of [ -8, 8] C. f ( ) oes not eist D. f () is not efine for < 0. E. f (0) oes not eist. If f (a) = f (b) an f () is continuous on [a, b], then A. f () must be ientically zero B. f( ) may be ifferent from zero for all on [a,b] C. there eists at least one number c, a < c< b, such that f( c) 0 D. f ( ) must eist for every on (a, b) E. none of the preceing is true. Fin the value of c that satisfies the Mean Value Theorem for f ( ) 4 on the interval [-, ]. A. B. C. 0 D. 4 E. None of these. 4. Fin the number that satisfies the MVT on the given interval or state why the theorem oes not apply. 5 a) f ( ) on [0, ] b) f( ) on [, 5] ( ) c) g( ) on [, ] ) h( ) ( ) on [, 9] 00 #9: Let f be efine by f ln change of f at cis the same as the average rate of change of f over,4? ( A) B.44 C.64 D.4 E.45. What is the value of c for which the instantaneous rate of

14 HW MVT Write the efinition of continuity. ) ) ) Write mathematical notation for ifferentiability: State the two prerequisite conitions that must be etermine before the Mean Value Theorem can be applie. ) ) What two calculations must be etermine before making a conclusion using the Mean Value Theorem. ) ) Rea questions #-4. If the function satisfies the hypotheses of the Mean Value Theorem, then solve for the value of c that satisfies the conclusion of the Mean Value Theorem. Otherwise, tell why it fails to meet the conitions of the Mean Value Theorem.. Given f 4 5, fin all values, c, in the interval [,4]. f, fin all values, c, in the interval [-,].. Given 4. Given f, fin all values, c, in the interval [-,]. 4. Given f, fin all values, c, in the interval [-8,8]. 4