MATH137 Chapter 1-3: Limits, Tangent Slope, Differentiation Rules

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sayheyfei

24 Oct 2016

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University of Waterloo

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Mathematics

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MATH137

Professor

Ken Giuliani

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A function is a rule that assigns to each element x in a set d exactly one element, called f(x), in a set e. The vertical line test: a curve in the xy-plane is the graph of a function of x if and only if no vertical. Symmetry line intersects the curve more than once: if (cid:1858)(cid:4666) (cid:1876)(cid:4667)=(cid:1858)(cid:4666)(cid:1876)(cid:4667), then the graph is even, if (cid:1858)(cid:4666)(cid:1876)(cid:4667)= (cid:1858)(cid:4666)(cid:1876)(cid:4667), then the graph is odd. 1. 2 mathematical models: a catalog of essential functions. Trigonometric functions: if it can be constructed by algebraic operations starting with polynomials. Combination of functions ((cid:1858)(cid:1859))(cid:4666)(cid:1876)(cid:4667)=(cid:1858)(cid:4666)(cid:1876)(cid:4667)(cid:1859)(cid:4666)(cid:1876)(cid:4667) (cid:4666)(cid:1858) (cid:1859)(cid:4667)(cid:4666)(cid:1876)(cid:4667)=(cid:1858)((cid:1859)(cid:4666)(cid:1876)(cid:4667)) (cid:1854)(cid:3051)+(cid:3052)=(cid:1854)(cid:3051)(cid:1854)(cid:3052) (cid:1854)(cid:3051) (cid:3052)=(cid:1854)(cid:3051)(cid:1854)(cid:3052) (cid:4666)(cid:1854)(cid:3051)(cid:4667)(cid:3052)=(cid:1854)(cid:3051)(cid:3052) (cid:4666)(cid:1853)(cid:1854)(cid:4667)(cid:3051)=(cid:1853)(cid:3051)(cid:1854)(cid:3051) A function f is called a one-to-one function if it never takes on the same value. Let f be a one-to-one function with domain a and range b twice, that is, To express (cid:1858) (cid:2869) as a function of x, interchange x and y.

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Sketch using these guidelines

pul aill al thes information together to sketch graphs that reveal the important You might ask: Why don't we just use a graphing calculator or computer to graph a It's true that modern technology is capable of producing very accurate graphs. But K 21+1+2 features of functions curve? Why do we need to use calculus? even the best graphing devices have to be used intelligently. It is easy to arrive at a misleading graph, or to miss important details of a curve, when relying solely on tech- nology. (See "Graphing Calculators and Computers" at www.stewartcalcelus.com, espe 4 discover the most interesting aspects of graphs and in many cases to and minimum points and inflection points exocily instead of approximately cially Examples 1, 3.4, and 5. See also Section 4.6.) The use of calculus enables us to calculate maximum FIGURE 1 For instance, Figure I shows the gr aph of /(x)-è¨-21x, + 18x + 2.At first glance it seems reasonable: It has the same shape as cubic curves like y x and it appears to have no maximum or minimum point. But if you compute the derivative, you will see that there is a maximum when x-0.75 and a minimum when x 1. Indeed if we zoom in to this portion of the graph, we see that behavior exhibited in Pigure 2 Without calculus, we could easily have overlooked it. 6a, FIGURE 2 In the next section we will graph functions by using the interaction between calculus and graphing devices. In this section we draw graphs by first considering the foilowing information. We don't assume that you have a graphing device, but if you do have one you should use it as a check on your work. â Guidelines for Sketching a Curve every item is relevant to every function. (For instance, a to make a sketch that displays the most important aspects of the function. The following checklist is intended as a guide to sketching a curve y- f(x) by hand. Not given curve might not have an asymptote or possess symmetry.) But the guidelines provide all the information you need A. Domain It's often useful to start by determining the domain D of f, that is, the set B. Intercepts The y-intercept is f(o) and this tells us where the curve intersects the of values of x for which f(x) is defined y-axis. To find the x-intercepts, we set y-0and solve for xã(You can omit this step if the equation is difficult to solve.) ry (a) Even fusction: reflectional symmetry all x in D, that is, the equation of the curve is unchanged when x is replacea by -x, then f is an even function and the curve is symmetric about the y-axis. This means that our work is cut in half. If we know what the curve looks like for 0, then we need only reflect about the y-axis to obtain the com- plete curve [see Figure 3(a). Here are some examples: y-ey-ry x). and for all x in D, then f is an odd function and the curve is symmetric about the origin. Again we can obtain the complete curve if we know what it looks like for x0. [Rotate 180Â° about the origin; see Figure 3(b).] Some b) Oud function rotational symmetry FIGURE 3 simple examples of odd functions are y-xy-x'y - and y-sin

5. (18 pts) Circle True if the statement is ALWAYS true; circle False otherwise. No explanations are required. Let f be a one-to-one function and f its inverse. (a) If the point (2, 5) lies on the graph off, then the point (5,2) lies on the graph of f. True False f (2) = 5 f-1 (5) = 2 (b) sin-1 (T) = 0 True False Since >1, ris not in the domain of sin-?; | or sin-1 (7) #0, since sin (0) = 0 # Iff and g are two functions defined on (-1, 1), and if (c) limg (x) = 0, then it must be true that lim[ f(x) g (x)] = 0. True False Let f (x) = 1 / x, if x #0, and f (0) = 4 and g(x) = x. Then lim g (x) = lim (x) = 0, but Lim ( f (x) g(x)] = lim [() x] = 18 0. Iffis continuous on (-1, 1), and if f (0) = 10 and lim g (x) = 2, (a) True False then lin (= 5. Because lim : (*) - Limx-of (*) .f(0) = 20 = 5 If f is continuous on [1, 3], and if f(1) = 0 and f(3) = 4, then the equation f(x) = (e) has a solution in (1, 3). True False Since f continuous on [1, 3), f(1) = 0 < < 4 = f (3), by the Intermidiate Value Theorem there exists a cin (1, 3) such that f (c) = . (f) Let f be a positive function with vertical asymptote x = 5. Then lim f (x) = +0. True False Let f (x) = 1, if x < 5, and f (x) ==, ifx>5. Then lim f (x) = +00, and therefore, X = 5 is a vertical asymptote. 5 - X So, f is a positive function with vertical asymptote x = 5 but lim f (x) + +co, since lim f (x) = + 0 + lim f (x) = 1, which means that lim f (x) does not exist.

MATH137 Chapter 1-3: Limits, Tangent Slope, Differentiation Rules

MATH137 Full Course Notes

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