MATH 113: ELEMENTARY CALCULUS

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1 MATH 3: ELEMENTARY CALCULUS Please check zhiyongz for notes updation! 6. Rates of Change and Limits A fundamental philosophical truth is that everything changes. In physics, the change in position is known as velocity or speed. In economics, the change in price is known as inflation. In business, the change in costs is sometimes known as trend. In mathematics, the change in values of a function is known as the derivative(limit of some ratio). But to understand the derivative or the limit, which will measure instantaneous change, you need to to first be comfortable with average change over some intervals. 6.. Average speed. Example. Calculating If you travel 200 miles in four hours, what is your average speed? What about 75 miles in one and a half hours? Questions to ponder: distance travelled Average speed = total time Do these calculations necessarily mean you went 50mph for four hours? Example 2. A ball which is dropped from the top of the Tower of Pisa has traveled down y = 6t 2 feet after t seconds.. What is its average speed over the first two seconds? 2. From second to second 2? 3. See Figure. Find the speed of the ball at t = and t = Calculating slopes of secant lines to a curve. Next we look at what at first appears to be unrelated to dropping a ball. Definition 3. A secant line goes through two points on the graph of the function. In symbols, it is a line through (x, f(x )) and (x 2, f(x 2 )) for some x and x 2. Example 4. Find the secant lines to the graph of f(x) = 6x 2 through the points with: x = 0, x 2 = 3, x = 0, x 2 = 5, x = 2, x 2 = Formula for average rate of change. Definition 5. In general, the average rate of change of some function f(x) as x varies between values x and x 2 is y x = f(x 2) f(x ) = f(x + h) f(x ), h = x 2 x 0 x 2 x h Expected: instantaneous rates = limiting values of average rates Limits. Definition 6. Let f(x) be defined on an open interval about except possibly at itself. If f(x) gets arbitrarily close to L (as close to L as we like) for all x sufficiently close to we say that approaches the limit L as x approaches lim x x 0 f(x) = L Note: x is sufficiently close to x 0, but x x 0. Example 7. Find Limits by substitution. 4 = x 2 x = x 3 (5x 3) = x 2 ( x 3 x 2 x + ) = { 2x +, x Example 8. Let f(x) = 2, x = find lim x f(x) =?

2 2 MATH 3: ELEMENTARY CALCULUS Figure. from Textbook Page 76 Example 9. Find 2x + 2 lim x x +. Note: The limit does NOT depend on how the function is defined at x 0 Example 0. lim sin( ) does NOT exist. x 0 x Example. lim x 0 x does NOT exist. 7.. the Limit Laws. 7. Calculating Limits Using the Limit Laws Property. Properties of limit.. lim x 3 k = k 2. lim x c x = c Example 2. Find Limits. Note: Whenever f(x) is an algebraic combination of polynomials and trigonometric functions for which f(x 0 ) is defined.. lim x c (x 3 4x 2 3) = x 4 + x 2 2. lim x c x 2 = x2 3 = lim x 2

3 MATH 3: ELEMENTARY CALCULUS 3 Figure 2. from Textbook Page Limits with cancellation. Question 3. Find the limit. x 2 x 2 x 2 x 2 x 0 x x 2 x 2 x 4 6 x 2 x x x Figure 3. from Textbook Page 84-85

4 4 MATH 3: ELEMENTARY CALCULUS x 3 x 2 9 x Theory and Examples. Example 4. If f(x) f(x) lim x 2 x 2 =, find (a) lim f(x) =, and (b) lim x 2 x 2 x =. f(x) Example 5. If lim x 0 x 2 =, find (a) lim f(x) = x Sandwich Theorem and Monotone property., and (b) lim x 0 f(x) x =. Example 6. Given that Figure 4. from Textbook Page x2 4 u(x) + x2 2 for all x 0, find lim x 0 u(x), no matter how complicated u is. Example 7. Find the limits.. lim sin x = x 0 2. lim cos x = x 0 3. If lim f(x) = 0, then lim f(x) = 0 x 0 x 0 8. One-Sided Limits and Limits at Infinity 8.. One-Sided Limits. Definition. One-side limit Figure 5. from Textbook Page x x f(x) denotes the limit of f(x) as x approaches x 0 from the left. 0 x x + f(x) denotes the limit of f(x) as x approaches x 0 from the right. 0

5 MATH 3: ELEMENTARY CALCULUS 5 Theorem 2. One-Sided Limits and Limits lim x x0 f(x) = L lim x x f(x) = lim 0 x x + 0 Example 3. One-sided limits from graph above. f(x) = L Example 4. Find the following limits. { 2x 3, x <. Given f(x) = consider the existence of lim x f(x) x, x 2. lim x 2 x 2 { x 2 +, x Example 5. Let f(x) = x, x >. Find lim x + f(x), lim x f(x) 2. Does lim x f(x) exists? 3. Sketch the graph. An important limit involving sin θ θ : Theorem 6. lim x 0 sin x x = Example 7. Find the following limits.. lim x 0 cos x x 2. lim x 0 tan 2x 5x (Just a remind:. When limit involving ratio of trigonometric functions, please consider Theorem Convert trigonometric functions in terms of sin, cos to evaluate.) 8.2. Limit as x approaches or. Definition 8. Limit at ± lim x f(x) = L if the value of f(x) is arbitrily close to L whenever x is sufficient large. lim x f(x) = L if the value of f(x) is arbitrily close to L whenever x is sufficient small. Example 9. lim x x = 0 lim x x = Horizontal Asymptotes. Definition 0. y = a is a horizontal asymptote of the graph of f(x) lim x f(x) = a or lim x f(x) = a A line having the property that the distance between this line and the graph approaches 0 as a point on the graph moves increasingly far from the origin. Example. Evaluate x e x + x e x + Property 2. Limits of exponential functions at infinity a m/n = ( n a) m +, a > lim x + ax =, a = 0, a < +, a < lim x ax =, a = 0, a > To remember which is which, it is sufficient to use 2 for a > and /2 for 0 < a <, and just let x run through positive integers as it goes to +. Likewise, it is sufficient to use 2 for a > and /2 for 0 < a <, and just let x run through negative integers as it goes to.

6 6 MATH 3: ELEMENTARY CALCULUS Example 3. What do you think of it? x x 9 3 x = The hazard here is that is not a number that we can do arithmetic with in the normal way. Dont even try it. So we cant really just plug in to the expression to see what we get. On the other hand, what we really mean anyway is not that x becomes infinite in some mystical sense, but rather that it just gets larger and larger. In this context, the crucial observation is that, as x gets larger and larger, /x gets smaller and smaller (going to 0). Thus, just based on what we want this all to mean, x x x = 0 = 0 x x 2 = 0 x This is the essential idea for evaluating simple kinds of limits as x : rearrange the whole thing so that everything is expressed in terms of /x instead of x, and then realize that lim is equivalent to lim x x 0 So, in the example above, divide numerator and denominator both by the largest power of x appearing anywhere: The point is that we called /x by a new name, y, and rewrote the original limit as x as a limit as y 0. Since 0 is a genuine number that we can do arithmetic with, this brought us back to ordinary everyday arithmetic. Of course, it was necessary to rewrite the thing we were taking the limit of in terms of /x (renamed y). Notice that this is an example of a situation where we used the letter y for something other than the name or value of the vertical coordinate. Question 4. Find the limit. x 2 x x 2 x x x 0 5x Oblique Asymptotes. If the degree of the numerator of a rational function is one greater than the degree of the denominator, the graph has an oblique (slanted) asymptote. We find an equation for the asymptote by dividing numerator by denominator to express f as a linear function plus a remainder that goes to zero as x ± Example 5. Find the oblique asymptote for the graph of y = x2 + x. Example. One-Sided Infinite Limits Find lim x + x and lim x x Example 2. Two-Sided Infinite Limits Discuss the behavior of a. f(x) = x 2 near x = 0 b. f(x) = near x = 3 (x + 3) 2 9. Infinite Limits and Vertical Asymptotes Example 3. Find the following limits.(rational Functions Can Behave in Various Ways Near Zeros of Their Denominators) a. lim x 2 (x 2) 2 b. lim x 2 x 2 c. lim x 2 + x 3 x 3 d. lim x 2 x 3 e. lim x 2 f. lim x 2 2 x (x 2) 3

7 MATH 3: ELEMENTARY CALCULUS 7 Figure 6. from Textbook Page 5-6 Figure 7. from Textbook Page 8

8 8 MATH 3: ELEMENTARY CALCULUS Definition 4. Vertical Asymptote Example 5. Find asymptotes of the graph of a.f(x) = 8 b. f(x) = x2 3 2x 4 Figure 8. from Textbook Page Continuity Definition. Continuity Interior point: A function f(x) is continuous at an interior point c of its domain if lim f(x) = f(c) x c Endpoint: A function is continuous at a left endpoint a or is continuous at a right endpoint b of its domain if lim f(x) = f(a), lim f(x) = f(b), respectively. x a + x b Note 2. For continuity test:. f is define at x 0 2. lim f(x) exist. x x 0 3. this limit is equal to f(x 0 ). Example 3. Examples of discontinuous functions. { x, x 0 a. f(x) = Removable discontinuity. Redefine f(0) = 0, then f is continuous., x = 0 { x, x 0 b. f(x) = Jump discontinuity. x +, x > 0 c. f(x) =. Infinite discontinuity at x = 0. x2 d. f(x) = sin 2π. Oscillating discontinuity at x = 0. x Note: f(x) is continuous on interval I f(x) is continuous at every point of interval I. Example 4. By Theorem 9, Polynomial, rational functions and absolute value functions are continuous.

9 MATH 3: ELEMENTARY CALCULUS 9 Example { 5. Show function f is continuous on R. x, x 0 f(x) = x 2, x > 0 Example 6. Find the values of a, b such that function f is continuous on R. x, x < f(x) = ax 2 + b, x < 3 2x, x > Continuous extension at a point. Example 7. Show that f(x) = x2 + x 6 x 2 has a continuous extension to x = 2, and find the extension. 4 Example 8. Show that f(x) = sin x has a continuous extension to x = 0, and find the extension. x Figure 9. from Textbook Page Example 9. Root finding: Show that f(x) = x 3 + x 5 has at least one root... Tangent slope.. Tangents and Derivatives Riview. Secant line to f through (a, f(a)) and (a + h, f(a + h)) has the slope m h = f(a + h) f(a) a + h a Definition 2. The tangent line to f at (x 0, f(x 0 )) has slope = f(a + h) f(a). h provided the limit exist. f(x 0 + h) f(x 0 ) m = lim, h 0 h

10 0 MATH 3: ELEMENTARY CALCULUS Example 3. Find the equation of tangent line to f(x) = x 2 + at (, 2). Example 4. Let f(x) = x. Show that there is no tangent line to f(x) at (0, 0). Example 5. Slope and tangent to f(x) =, x 0 at (, 2). x a. find the slope of the curve at x = a 0 b. where does the slope equal 4? c. what happens to the tangent to the curve at the point (a, ) as a changes? a.2. Formula for instantaneous rate of change. We would see similarities in trying to compute average and instantaneous differences regardless of what our functions are measuring. Definition 6. The derivative of f(x) with respect to x at x 0 is the function f f(x 0 + h) f(x 0 ) (x) x=x0 = lim. h 0 h We will talk about lim, which means the limit, later. For now, we will adopt a working notion that we may compute this for smaller and smaller h; if all of those computations seem to approach a single number, for now we will call that the limit. Note 7. Finding the Tangent to the Curve y = f(x) at (x 0, y 0 ),.Calculate f(x 0 ) and f(x 0 + h) f(x 0 + h) f(x 0 ) 2. Calculate the slope m = lim h 0 h 3. If the limit exists, find the tangent line as y = y 0 + m(x x 0 ). Example 8. A ball which is dropped from the top of the Tower of Pisa has traveled down y = 6t 2 feet after t seconds. Find the speed of the ball at t = and t = 2.