Limits and Derivatives in Calculus

Lecture 1: Limits, Derivatives, the Product Rule, the Quotient Rule, and the Chain Rule

Part I: Limits and Continuity

Objectives Know what left limits, right limits, and limits are Know how to compute simple limits Know what it means for a function to be continuous Know how sin(?) and cos(?) behave as ? 0. Corresponding sections in Simmons: 2.5, 2.6

What is a limit? A limit is what happens when you get closer and closer to a point without actually reaching it. Example: If ?(?) = 2? then as ? 1, ? ? 2. We write this as lim ? 1? ? = 2. x 0 .9 .99 .999 .9999 f(x) 0 1.8 1.98 1.998 1.9998

Why are limits useful? Many functions are not defined at a point but are well-behaved nearby. ?2 1 ? 1 then ? 1 is undefined. However, as ? 1, ? ? 2, so lim ? 1? ? = 2. Example: If ?(?) = 4 3 2 1 x 0 .9 .99 .999 .9999 f(x) 0 -1 -2 -3 -4 f(x) 0 1.9 1.99 1.999 1.9999 -2 -1 x0 1 2 3 4 -4 -3

Left Limits and Right Limits ? ?. ? 0 is undefined. As ? 0 , ?(?) = 1 Consider ?(?) = 4 3 x -1 -.1 -.01 -.001 -.0001 2 f(x) -1 -1 -1 -1 -1 1 As ? 0+, ?(?) = 1 0 -1 -2 -3 -4 f(x) x 1 .1 .01 .001 .0001 f(x) 1 1 1 1 1 -2 -3 -1 0 1 2 3 4 -4 x We write this as lim ? 0 ? ? = 1, lim ? 0+?(?) = 1

Limit Definition Summary We say that lim ? We say that lim ?+ If lim ? ? ? ? = lim matter which side x approaches a from) then we say that lim ? ?? ? = ? ? ? ? ? = ? if ? ? ? as ? ? ?+? ? = ? if ? ? ? as ? ? ?+? ? = ?(i.e. it doesn t

Nonexistence of Limits Limits can fail to exist in several ways 1. lim ? ? ? ? or lim Example: sin ? oscillates rapidly between 0 and 1 as ? 0+ (or 0 ). Thus, lim DNE (does not exist) Example: 1 write this as lim ? 0+ 2. lim ? ? ? ? and lim have different values. Ex: ? ? = ? ?+? ? may not exist. 1 1 ? ? 0+sin ? gets larger and larger as ? 0+. We 1 ?= ? ?+? ? may both exist but ? |?| near ? = 0

Computing Limits To compute lim ? ?? ? : If nothing special happens at ? = ?, just compute ? ? . Example: lim ? 2(3? 1) = 5 If plugging in ? = ? gives 0 cancelled when ? ?. Example: 0, factors can often be ? 2(?2 4 ? 2((? 2)(?+2) lim ? 2) = lim ) = lim ? 2(? + 2) = 4 ? 2

Computing Limits Continued ?+?=?2 ?2 Useful trick: ? ? = ? ? ?+? ?+? ?+1 1 ? ? + 1 1 ? ( ? + 1 + 1)=1 Example: What is lim ? ? 0 ? + 1 1 ? ? + 1 + 1 lim ? 0 = lim ? 0 ? + 1 + 1 ? 1 = lim ? 0 ?( ? + 1 + 1)= lim 2 ? 0

Limits at Infinity We can also consider what happens when ? or ? . Example: Consider ? ? = ? 1 ? write this as lim ? ? = 1 1 ?. As x (or ), ? ? 1. We ? 1 = 1

Computing Limits at General strategy : figure out the largest terms and ignore everything else 3?2 ? 4?2+2? 5 , as ? only Example: If ? ? = the 3?2 in the numerator and the 4?2 will really matter, so lim ? ? ? = 3 4

Growth Rates as ? As ? , 1 ln? ?? ?? as long as ? > 0 and ? > 1 Examples: ??? lim ? ?= 0 2? 5?100= lim ?

Limit Laws If lim ? ?? ? = L and lim lim ? ?(? ? + ?(?)) = L + M lim ? ?(? ? ?(?)) = L M lim ? ?(? ? ?(?)) = LM ? ?(?(?) Etc. ? ?? ? = ? then: ? ? (if ? 0) lim ?(?)) =

Continuity Definition: ? ? is continuous at a if both ? ? and lim ? ?? ? exist and are equal. Note: Polynomials are always continuous everywhere. Most functions we will be working with are continuous almost everywhere.

Discontinuous functions ? ? may fail to be continuous at ? = ? because: 1. lim ? ?? ? or ? ? does not exist. Example: If ? ? = ? then lim exist. ?2 1 ? 1 then ? 1 is undefined. 2. lim ? ?? ? or ? ? both exist but have different values. Example: If ? ? = ? ? then lim but ? 1 = 0 ? 0? ? does not Example: If ? ? = ? 1? ? = 1

Two Trigonometric Limits How do sin(?) and cos(?) behave as ? 0? Consider lim ? 0 directly, but if we think about it ???? ?. We can t cancel ???? and ? x 1 1 sin(x) x x sin(x) x As ? 0, ???? looks more and more like ?, so lim ? 0 ? ???? = 1.

Two Trigonometric Limits (cont.) How about cos(?)? Clearly, lim ? 0???? = 1, but 1 ???? ? consider lim . ? 0 1 ???? ? ???2(?) ?(1+ ????)= lim (1 ????)(1+ ????) ? (1+ ????) ???? ? lim ? 0 lim ? 0 lim ? 0 = lim ? 0 = ???? 1+ ????= 0 ? 0 ?2 2. In fact, for small x, ???? 1

Part II: Derivatives

Objectives: Know what a derivative is Know how to compute simple derivatives from the definition of a derivative Know some basic facts about continuous and differentiable functions. Corresponding Sections in Simmons: 2.3,2.6

What is a derivative? The derivative ? ? of a function ?(?) says how fast ?(?) changes as ? changes. Visually, ? ? is the slope of ?(?) at ?. 4 1 4?2 Example: If ? ? = then ? 2 = 1 because the slope of ?(?) at ? = 2 is 1. We can see this by looking at the tangent line to ?(?) at ? = 2. 3 2 1 0 -1 -2 -3 -4 f(x) -2 -1 0 1 2 3 4 -3 -4 x

Why are derivatives useful? Tells us how quickly something is changing. In physics: velocity is the derivative of position and acceleration is the derivative of velocity (with respect to time). Optimization: Derivatives are crucial for finding the minimum or maximum of functions. And much much more.

Computing derivatives To compute the slope of a line, we take ? ? (rise/run) We could try to do the same thing with a function, taking ? ?+ ? ?(?) ?+ ? ? ? Unfortunately, the slope of f(x) can change with x, so we get the average slope of f(x) over the interval [?,? + ?] rather than the exact slope of f(x) at x. However, if we make ? smaller and smaller, the slope of f(x) varies less and less in [?,? + ?] and we get a better and better estimate. =? ?+ ? ?(?) 4 3 2 1 ? ? =?3 0 -1 -2 -3 -4 16 -2 -1 0 1 2 3 4 -3 -4 x

Derivative Definition and Examples We accomplish this by taking the limit as ? 0. ? ?+ ? ?(?) ? Definition: f x = lim ? 0 If f (x) exists then we say that f is differentiable at x Example: If ? ? = 3? + 4 then f x = lim ? 0 Example: If ? ? = ?2 then f x = (?+ ?)2 ?2 ? = 2? ?+( ?)2 ? = lim 3 ?+ ? +4 (3?+4) ? 3 ? ?=3 = lim ? 0 lim ? 0 lim ? 0 ? 02? + ? =2?

Leibniz Notation So far, we have written the derivative of a function f as f . Another notation, devised by Leibniz, is ?? ??. Warning: ?? on their own. Advantages of Leibniz notation: Emphasizes how the derivative is computed. ?? ?? is a single function. ?? and ?? do not have values ? ? ??= lim ? 0 Makes it easier to express the product rule, quotient rule, and chain rule Disadvantage of Leibniz notation: ?? ???=2 or ?? ???=2to write the Need clumsy notation like derivative of a function at a particular point.

Differentiable Implies Continuous Restatement of continuity: f is continuous at x if and only if ? ? exists and lim ? 0 ? = 0 where ? = ? ? + ? ? ? . ? ? exists f is differentiable ? (?) = lim ? 0 If f is differentiable at x then ? ? lim ? 0 ? = ? ? 0 = 0 lim ? 0 ? = lim Thus, differentiability implies continuity Warning: The converse is false. Not all continuous functions are differentiable! ? 0

The Extreme Value Theorem Theorem: If f is continuous on a closed interval ?,? then f takes a maximum and minimum value on ?,? . More precisely, there are ?,? [?,?] such that for all ? [?,?], f c ? ? ?(?) The extreme value theorem seems obvious, but it is not so easy to prove it rigorously The extreme value theorem depends on all of its hypotheses. The statement is not true if f is not continuous or we consider the open interval (a,b) rather than the closed interval ?,? .

The Intermediate Value Theorem Theorem: If f is continuous on a closed interval ?,? then f assumes every value between ?(?) and ?(?). More precisely, for any K such that K is strictly between ?(?) and ? ? , there is a ? (?,?) such that ? ? = ?. Corollary: If f is continuous and ?(?) and ? ? have opposite signs then there is a ? (?,?) such that ? ? = 0.

The Mean Value Theorem Theorem: If f is continuous on a closed interval ?,? and differentiable on (?,?) then there is a ? (?,?) such that ? ? = ? ? Note that this is the slope of the line from (?,?(?)) to (?,?(?)) ? ? ?(?) 4 3 1 4?2then Example: If ?(?) = then if we take ? = 0 and ? = 4, the mean value theorem tells us that there is a ? (0,4) such that ? ? = ? 4 ?(0) 4 0 = 1. This is indeed true as ? 2 = 1 2 1 0 -1 -2 -3 -4 f(x) -2 -1 0 1 2 3 4 -3 -4 x

Average and Instantaneous Speed The idea behind the derivative is that if we look at smaller and smaller intervals (taking the limit as the length of the interval approaches 0), the average speed over the interval approaches the instantaneous speed. The mean value theorem says that over any interval, there is some point where the instantaneous speed matches the average speed. Warning: These statements are only true for differentiable position functions. Otherwise, weird stuff can happen.

Other Consequences of the Mean Value Theorem If ? ? > 0 on an interval then f is increasing on that interval (increasing means that ? < ? implies that ? ? < ?(?)). If ? ? < 0 on an interval then f is decreasing on that interval. If ? ? = 0 on an interval then f is constant on that interval.

Part III: Rules for Derivatives

Objectives: Know the derivatives of ??, sin(?), cos(?) Know the product rule, quotient rule, and chain rule and be able to use them to compute sums, products, quotients, and compositions of these functions. Corresponding sections in Simmons: 3.1,3.2,3.3, 3.4

Derivative of ?? ? ?( ?)??? ? ? For nonnegative integers n, ? + ??= ?=0 Examples: ? + ?2= ?2+ 2 ? ? + ( ?)2 ? + ?3= ?3+ 3 ? ?2+ 3( ?)2? + ( ?)3 ? + ?4= ?4+ 4 ? ?3+ 6( ?)2?2+ 4( ?)3? + ( ?)4 =(??+? ? ?? 1+ ?2( )) ?? ? ?+ ?? ?? ? = lim If ? ? = ?? then ? ? = ??? 1 This holds for all n, not just nonnegative integers! We ll prove this for rational numbers later using implicit differentiation. ?+ ?? ?? ? = ??? 1+ ( ?)( ) ? 0??? 1+ ? = ??? 1 lim ? 0

Derivative of sin(?) sin ?+ ? sin(?) ? sin ?+ ? sin(?) ? sin ?+ ? sin(?) =sin ? cos ? +cos ? sin ? sin(?) ? (cos ? 1) ? sin ? ? = sin ? + cos ? (cos ? 1) ? sin ? ? lim ? 0 = sin ? lim + cos ? lim ? ? 0 ? 0 sin ? ? Recall that lim = 1 ? 0 (cos ? 1) ? Recall that lim = 0 ? 0 sin ?+ ? sin(?) ? lim ? 0 = sin ? 0 + cos ? 1 = cos(?) If ? ? = sin ? then ? ? = cos ?

Derivative of cos(?) Following similar reasoning, if ? ? = cos ? then ? ? = sin ?

Derivatives of Sums and Differences ?(?+?) ?? ?(? ?) ?? This seems intuitive, but let s check the first equation to be sure. Take ? = ? ? + ? ?(?) =?? ??+?? =?? ?? ?? ?? ?? ?(?+?) ?? lim ? 0 (?+?) ? ? ?=?? ?+ ? ? = lim ? 0 ?+ lim = lim ? 0 ??+?? = ? ?? ? 0

The Product Rule What is ?(??) ??? Warning: ?(??) ?? ?? ?? ?? ?? ?? = ? + ? ? + ? ?? ?? = ? ? + ? ? + ? ? ? ?+? ?+ ? ? ? ?(??) ?? (??) ? = lim ? 0 = lim ? 0 ?(??) ?? ? ?+ ? lim ? ?+ lim ? ? ? = ? lim ? 0 ??+ ??? ? 0 ? 0 ?(??) ?? = ??? ??

The Quotient Rule ? ? ? What is ??? ? ? ?? ?? ?? ?? ?=??+? ? ?? ? ? ? ?+ ? ? Warning: ?? ? ? ?+ ? ?+ ? ? =? ? ? ? ? ?+ ? = ? ? ? ? ? ? ? ? ? ?+ ? ? ? ? ? lim ? ? ? ? lim ? 0 lim ? 0?(?+ ?) ? 0 ??= lim ?= lim = ? 0 ? 0 ? ? ??? ?? ??? ?2 ? ?? ??=

The Chain Rule What is ? ??(?(?)) where ? is a function of ?? Chain rule: ?? ?? Example: If ? ? = 1 + ?2 and ? ? = ?? ??=?? ??=?? ?? ?? 1 + ?2 then taking ? = ?, 1 2 ? 2? = ?? ?? ? ??= 1 + ?2

Reasoning behind the Chain Rule Proof of the chain rule (has a flaw, but the intuition is right!): ?? ??= lim ? 0 ? 0 Technical issue: What if ? = 0? Fix: ? =?? ?? ? + ? ? where ? 0 as ? 0 (and thus as ? 0) Now ? ?=?? ? 0 ? ?= lim ? ? ? ? ? lim ? ?=?? ?? ?? ?= lim ?? ? 0 ? 0 ? ?+ lim ? ?=?? ?? ?? lim ? 0 ??lim ? 0? lim ?? ? 0