Implicit Differentiation, Inverse Trigonometric Functions, Higher Derivatives

Lecture 2: Implicit Differentiation, Inverse Trigonometric Functions, Higher Derivatives, Curve Sketching, and Min/Max Problems

Part I: Exponentials, Implicit Differentiation, and Inverse Trigonometric Functions

Objectives Know the derivatives of ??and ln?. Know implicit differentiation and how to use it to find the derivative of exponential functions and inverse trigonometric functions. Corresponding Sections in Simmons: 3.5, 8.3,8.4, 9.5

What is ?? ? (1 +1 ?)? ? = lim Most important property of ? is that ? ????= ??

Derivative of ?? If ? ? = ??then ??+ ? ?? ? ? ? 1 ? ? ? = lim = ??lim ? 0 ? 0 To get ? ? = ? ? = ??, a must be the ? ? 1 ? number such that lim = 1 ? 0 1 This gives ? ? 1 ?, so ? (1 + ?) ? 1 ? = lim ? 0(1 + ?) ?= ?

Series for ?? ? (1 +1 When ?? is an integer, ??= lim ?)?? 2 3 (1 +1 When ? is extremely large, this is very close to 2 +(??)3 3 2 ?? ?!so ? = 1 + 1 + Note that the derivative of each term in the series is the previous term! 1 ?+??(?? 1) 1 ? +??(?? 1)(?? 2) 3 2 1 ? ?)??= 1 + ?? + 2 3 1 ?+(??)2 + = 1 + x +?2 2!+?3 1 ? 1 ? 1 + ?? 3!+ 2 1 2!+ 1 3!+ ??= ?=0

Differentiating Implicit Functions We like writing functions as ? = but sometimes this is difficult if not impossible Example: If we know that ?4?4+ ?2? = 1, ? clearly depends on ?, but it is extremely difficult to solve for ?. Fortunately, we don t need to solve for ? in terms of ? to find ?? ??. Just differentiate both sides with respect to ? and solve for ?? ??!

Example If ?4?4+ ?2? = 1, differentiating both sides with respect to ? gives 4?3?4+ 4?4?3?? ??+ 2?? + ?2?? ??= 0 Solving for ?? ??gives ?? ??= 4?3?4+ 2?? 4?4?3+ ?2

The Power Law Revisited This idea is often useful even when we can write ? in terms of ?. ? ?where p and q are nonzero Consider ? = ? integers. ??= ??so taking the derivative with respect to ? gives ??? 1?? ??= ??? 1. Solving for ?? ?? ??=??? 1 ??gives ? ??? ? ? 1 ??? 1=? ???=? ? ? ?=? ? ?

Derivative of ln(?) If ? = ln(?) then ??= ? so taking the derivative with respect to ? gives ???? ??= 1. Solving for ?? ??gives?? ??=1 1 ??= ?

Derivative of ?? If ? = ??then ln(y) = ???(?) so taking the derivative with respect to ? gives 1 ? ?? ??= ln ? + 1. Solving for ?? ?? ??= ln ? + 1 ? = ln ? + 1 ?? ??gives

Derivative of sin1? If ? = sin 1? then sin(y) = ? so taking the derivative with respect to ? gives cos(?)?? ??= 1. Solving for ?? ?? ??= ??gives 1 cos(?)= 1 1 1 sin(?)2= 1 ?2

Derivative of t??1? If ? = tan 1? then sin(?) cos(?)= ? so taking the derivative with respect to ? gives (1 +sin(?)2 cos(?)2)?? ??= 1. Solving for ?? ??gives ?? ??= 1 1 1 + tan(?)2= 1 + ?2

Part II: Higher Derivatives

The Second Derivative The derivative ? ? of a function ?(?) is itself a function and we can take its derivative. This gives us the second derivative ? (?) (in Leibniz notation this is written as ?2? ??2) This is the rate of change of the rate of change of ?.

The Second Derivative In physics, the second derivative of position is acceleration. ? ? = ? (?) Also described by statements like, The GDP growth for the United States slowed last year. Visually, ? ? describes how concave up or concave down ? is. ? ? =?3 16 , ? ? =3? 8 4 ? (?) > 0 3 2 1 ? ? = 0 0 -1 -2 -3 -4 ? (?) < 0 -2 -1 0 1 2 3 4 -3 -4 x

Higher Derivatives We can find higher derivatives as well. The nth derivative of ? is written as ??(x) or ??? These derivatives are less commonly seen in practice but can be useful for better and better approximations with Taylor series. Example: If ? ? = ?? then ??x = ?? for all ?. ???.

Part III: Curve Sketching

Objectives Be able to sketch functions, including their critical points, discontinuities, zeros, and asymptotes. Corresponding Sections in Simmons: 4.1,4.2

Critical Points Def: The critical points of a function ?(?) are the points where ? ? = 0 Fact: Any local maximum or minimum of ?(?) occurs at either a critical point, the boundary, or a point where ? (?) is discontinuous or does not exist. Warning: Critical points are not always local maximums or minimums (but they usually are)

Examples: 6 5 4 3 2 1 ? ? =?2 0 -1 -2 -3 -4 -5 -6 4 -6 -5 -4 -2 0 1 2 3 4 -3 -1 5 6 x The critical point (0,0) is a minimum

Examples: 6 5 4 3 2 1 ? ? =?3 0 -1 -2 -3 -4 -5 -6 16 -6 -5 -4 -2 0 1 2 3 4 -3 -1 5 6 x The critical point (0,0) is not a minimum or maximum

Examples: 6 5 4 3 2 1 ? ? = |?| 0 -1 -2 -3 -4 -5 -6 -6 -5 -4 -2 0 1 2 3 4 -3 -1 5 6 x (0,0) is a minimum but not a critical point.

Information for Sketching Functions To help sketch a function ?(?), determine: 1. The critical points of ?(?) as well as the points where ? (?) is DNE. 2. The sign of ? (?) between these points. 3. The zeros of ?(?) 4. The behavior of ?(?) at the boundary (could be ) 5. The behavior of ?(?) near points where ? (?) is DNE. 6. It is also very useful to have a general picture of ?(?) and plot points that are easily found.

Examples If? ? = ?3 3? then: 1. The critical points of ?(?) are ( 1,2) and (1, 2). ?(?) has no discontinuities. 2. ? ? = 3?2 3 is positive to the left and right of these critical points and is negative in between them. 3. The zeros of ?(?) are 3,0 , 0,0 , and ( 3,0). 4. ?(?) as ? and ? ? as ?

Examples: 6 5 4 3 2 1 ? ? = ?3 3? 0 -1 -2 -3 -4 -5 -6 -6 -5 -4 -2 0 1 2 3 4 -3 -1 5 6 x

Examples ?2+ 1 then: If? ? = 1. The only critical point of ?(?) is 0,1 and ?(?) has no discontinuities. ? ?2+1 is negative to the left of this critical point and positive to the right of it. 3. ?(?) has no zeros 4. ?(?) as ? . More specifically, ? ? |?| 0 as ? 2. ? ? =

Examples: 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 ?2+ 1 ? ? = -5 -6 -4 -2 0 1 2 3 4 -3 -1 5 x 6

Examples ? then: If? ? = ? +1 1. The critical points of ?(?) are 1, 2 and 1,2 . ?(?) has a discontinuity at ? = 0. 1 2. ? ? = 1 for 1 < ? < 1 (except at ? = 0), and positive for ? > 1 3. ?(?) has no zeros 4. ?(?) as ? and ? ? as ? . More specifically, ? ? ? 0 as ? 5. ? ? as ? 0 and ? ? as ? 0+ ?2 is positive for ? < 1, negative

Examples: 6 5 4 3 2 1 ? ? = x +1 0 -1 -2 -3 -4 -5 -6 ? -5 -6 -4 -2 0 1 2 3 4 -3 -1 5 x 6

Asymptotes and Inflection Points Asymptotes are lines which the functions approaches in some limit. In the previous two examples, the red lines were asymptotes for ?(?). Asymptotes can occur as vertical lines at discontinuities or as limits as ? . Inflection points are points where ?(?) changes from concave up to concave down. This usually (but not always) occurs when ? ? = 0

Part IV: Min/Max Problems

Objectives Know how to solve min/max problems by looking at the critical points and other points of interest Corresponding sections in Simmons: 4.3,4.4

Finding Absolute Minima/Maxima Recall: Any local maximum or minimum of ?(?) occurs at either a critical point, the boundary, or a point where ? (?) is discontinuous or does not exist. To find the absolute minimum or maximum of ?(?), we just need to find all of these points and compare the values of ?(?) at these points.

Example Let s say that we want to minimize or maximize the function y = 2 ? + 1 ? over the interval [0,1]. ? = ? have that 1 ?= 4(1 ?) = ?, giving ? =4 1 1 2 1 ? so solving for ? = 0 we 1 2 1 ? which implies that 5. When ? =4 1 4 5= point is (4 5, 5). The boundary points are (0,1) and (1,2) so ? is maximized at (4 and minimized at (0,1). 5, ? = 4 5+ 4 1 2 5+ 5= 5. The critical 5, 5)

Example Word Problem What is the minimum perimeter of a rectangle with area 25? w l Area is lw = 25 Perimeter is 2? + 2? ? =25 ? so we are trying to minimize P x = 2? +50 ? for l > 0 (l < 0 makes no sense here)

Example Word Problem Continued we are trying to minimize P = 2? +50 0 Solving for P l = 0 gives 2 50 gives l = 5 (we ignore the solution l = 5 because we are only looking at ? > 0) Looking at the boundaries, lim lim ? ? ? = . ? is minimized when w = ? = 5 and gets larger and larger as l 0+ or l . The answer is ? 5 = 20 ? for l > ?2= 0 which ? 0+? ? = and

Solving Word Problems To solve word problems, we often: 1. Draw a picture for the problem 2. Find equations for the relevant variables 3. Use these equations to re-express what we re trying to minimize/maximize as a function in one variable. 4. Minimize or maximize this function.

The second derivative test How can we tell if a critical point is a local maximum or a local minimum? If ? ? > 0, it s a local minimum If ? ? < 0, it s a local maximum Otherwise, the second derivative test is inconclusive.

Reflection What is the minimum length of a path from A to B that bounces off the mirror? A B a b ?1 ?2 x c-x We are trying to minimize ? ? = ?2+ ?2+ (? ?)2+?2

Reflection A B a b ?1 ?2 x c-x We are trying to minimize ? ? = ? ? = ?2+ ?2 cos ?1 cos ?2 = 0 so ?1= ?2 Can also solve this to get ? = Can check that ? ? > 0, so this is a minimum. ?2+ ?2+ ? (? ?)2+?2 ? ? ? ?2+ ?2= 0 ? ?+? ?

Refraction What is the minimum time it takes to go from A to B if we have speed ?? above and speed ?? below? A ? a ? c-x x ? b ? B We are trying to minimize ?2+ ?2 ?? (? ?)2+?2 ?? ? ? = +

Refraction A ? a ? c-x x ? b ? We are trying to minimize B ?2+ ?2 ?? ? (? ?)2+?2 ?? ? ? ? ?2+ ?2= 0 = 0 so sin(?) ? ? = + ? (?) = ?? ?2+ ?2 sin(?) ?? ?? sin ? sin(?)=?? ?? ??