Math Chapter 1 Essentials of Calculus by James Stewart Prepared by Jason Gaddis

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1 Math 23 - Chapter Essentials of Calculus by James Stewart Prepared by Jason Gaddis Chapter - Functions. - Functions and their representations Definition A function is a rule that assignes to each element x in a set A exactly one element, called f(x), in a set B. The set A is the domain of the function and f(x) is the value of f at x. The set of all f(x) is called the range. A symbol that represents an arbitrary number in the domain is called an independent variable. A symbol that represents an arbitrary number in the range is called a dependent variable. (Diagram a function f : A B). Note To find the domain of a function, we must first determine what the excluded values are. Common situations to consider are square roots, denominators of fractions, and logarithms. Example Find the domain of each function. (a) f(x) = x 4 (b) g(x) = 5x+4 x 2 +3x+2 Definition A graph of a function is the set of ordered pairs {(x, f(x)) x A}. This is how we visualize a function. (Diagram) Note To determine if a graph represents some function, we can utilize the vertical line test: a curve in the xy-plane is the graph of a function of x if and only if (iff) no vertical line intersects the curve more than once. Example Draw a graph that represents a function and one that does not. Note Read pg. 5 for a review of Piecewise functions. Definition If a function f satisfies f( x) = f(x) for every x in the domain, then f is an even function. If a function f satisfies f( x) = f(x) for every x in the domain, then f is an odd function. Example f(x) = x 2 is even and f(x) = x 3 is odd. Note An even function is symmetric with respect to the y-axis. An odd function is symmetric with respect to the origin. (Diagram with above functions). Example Determine whether the following functions is even, odd, or neither. f(x) = x2 x 4 +. Example Determine whether the following functions is even, odd, or neither. f(x) = x x. Definition A function f is called increasing on an interval I if f(x ) < f(x 2 ) whenever x < x 2 in I. A function f is called decreasing on I if f(x ) > f(x 2 ) whenever x < x 2 in I. (Diagram)

2 .2 - A Catalog of Essential Functions Definition If the graph of a function is a line, we say that it is a linear function. We can express it in slope-intercept form: y = f(x) = mx + b. Definition A function P is called a polynomial if P (x) = a n x n + a n x n + + a 2 x 2 + a x + a 0, where n is a nonnegative integer and the numbers a 0, a,..., a n are constants called coefficients. If a n 0, then the degree of the polynomial is n. Note The domain of any polynomial is R = (, ). Example Types of Polynomials A degree two polynomial is a quadratic function: P (x) = ax 2 + bx + c. If a > 0, the graph opens upward and if a < 0, the graph opens downward. (Diagram) A degree three polynomial is a cubic function: P (x) = ax 3 + b 2 x 2 + cx + d (see examples pg. 3). Definition A function of the form f(x) = x a, where a is a constant, is called a power function. Example Types of Power Functions a = n, where n is a positive integer. a = /n, where n is a positive integer. a =. Definition A rational function f is a ratio of two polynomials, f(x) = P (x) Q(x) Note The domain of a rational function consists of all values of x such that Q(x) 0. Remark The trigonometric functions, sine and consine, are also important. f(x) = sin(x) and g(x) = cos(x) is (, ) and they have the property that Note that the domain of These functions are periodic with period 2π. sin(x) and cos(x). Definition The exponential functions are the functions of the form f(x) = a x, where a is a positive constant. The logarithmic functions are the functions of the form f(x) = log a (x), where a is a positive constant. Note The exponential and logarithmic functions with the same base are inverses of each other. Transformations of Functions: Now that we have a catalog of basic functions, we wish to discover a variety of new functions by performing various transformations on the basic functions. Definition A translation is a vertical or horizontal shift by c units. y = f(x) y = f(x) + c (vertical shift up if c > 0 and down if c < 0). y = f(x) y = f(x + c) (horizontal shift right if c < 0 and left if c > 0). 2

3 Definition A reflection flips the graph over the x- or y-axis. y = f(x) y = f(x) reflects the graph about the x-axis. y = f(x) y = f( x) reflects the graph about the y-axis. Definition Stretching scales the image by a factor of c either vertically or horizontally y = f(x) y = cf(x) (vertical: stretches if c > and compresses if 0 < c < ) y = f(x) y = f(cx) (horizontal: stretches if c > and compresses if 0 < c < ) Example Let f(x) = sin x. Perform the following transformation and write the resulting function: shift up 2 and left stretch vertically by 2 units stretch horizontally by 3 units reflect over the x-axis Combinations of Functions: Now that we have a catalog of basic functions, we wish to discover a variety of new functions by performing various transformations on the basic functions. Note We can add, subtract, multiply, and divide function pointwise: (f ± g)(x) = f(x) ± g(x), (fg)(x) = f(x)g(x), ( ) f (x) = f(x) g g(x) Note If the domain of f is A and the domain of g is B, then the domain of f ± g and fg is A B. The domain of f/g is {x A B g(x) 0}. Definition Given two functions f and g, the composite function f g is defined by (f g)(x) = f(g(x)). Example If f(x) = x 2 and g(x) = /x, find the composite functions f g and g f. What are the domains of the compositions? Note In general f g g f. 3

4 .3 - The Limit of a Function Note If we look at a basic function like f(x) = x + 2, then it is easy to see that when x = 2, f(x) = 4. However, that is impossible when f(x) = x 2 x 2 4. This is why we study limits, we want to evaluate functions at values we may not otherwise be able to (including ). Note We define average velocity to be the ratio of change in position versus change in time. The instaneous velocity is found by taking the limit as the change in time goes to zero. The next example illustrates this. Example If an arrow is shot upward on the moon with a velocity of 58 m/s, its height in meters t seconds later is given by h(t) = 58t 0.83t 2. Find the average velocity over the given time intervals [, 2], [,.5] and [,.]. Definition We write lim f(x) = L and say the limit of f(x), as x approaches a, equals L if we can make the values of f(x) arbitrarily close to L by taking x to be sufficiently close to a but not equal to a. Example Guess the value of lim x x 2 2x x 2 x 2. Definition We write lim f(x) = L and say the left-hand limit of f(x) as x approaches a is equal to L if we can make the values of f(x) arbitrarily close to L by taking x to be sufficiently close to a and x less than a. Note The right-hand limit lim + f(x) = L is defined similarly. Note A limit exists if and only if the left- and right-hand limits exist and are equal. That is, lim f(x) = L if lim f(x) = L and lim f(x) = L. + Example (page 33, number 4) Note We now make the concept of arbitrarily close more precise. Definition Let f be a function defined on some open interval that contains the number a, except possibly a itself. Then we say that the limit of f(x) as x approaches a is L, and we write, lim f(x) = L if for every number ε > 0 there is a corresponding number δ > 0 such that if 0 < x a < δ then f(x) L < ε. Note (diagram on pg 32) This must work for all ε > 0, no matter how small. In fact, we are usually only concerned with ε very small. Example Prove that lim x 2 ( 2x + 3) = 2 using the ε δ definition of a limit (use diagram). If we set ε =., what is the corresponding δ value? 4

5 .4 - Calculating Limits Note In this section we learn rules for evaluating limits. We also learn some important limits that can be obtained by unconventional means. Prop Limit Laws Suppose that c is a constant and the limits exist. Then lim f(x) and lim g(x). lim [f(x) + g(x)] = lim f(x) + lim g(x) 2. lim [f(x) g(x)] = lim f(x) lim g(x) 3. lim [cf(x)] = c lim f(x) 4. lim [f(x)g(x)] = lim f(x) lim g(x) 5. lim f(x) g(x) = lim f(x) lim g(x) [ ] n lim f(x) 6. lim [f(x)] n = if lim g(x) 0 where n is a positive integer 7. lim c = c 8. lim x = a 9. lim x n = a n where n is a positive integer 0. lim n x = n a where n is a positive integer and a > 0 if n is even. lim n f(x) = n lim f(x) assuming lim f(x) > 0 if n is even Example Evaluate the following limits and justify each step (a) lim x4 + 2u + 6 x 2 (b) lim x 0 cos 4 x 5 2 x 3 Prop Direct Substitution Property If f is a polynomial or a rational function and a is in the domain of f, then lim f(x) = f(a). Note If f(x) = g(x) when x a, then lim f(x) = lim g(x), provided the limits exist. Example Evaluate the limit x 2 + 5x + 4 lim x 4 x 2 + 3x 4. Theorem lim f(x) = L if and only if lim + f(x) = L and lim f(x) = L. Example (pg 40, example 6 and 7) Show that lim x = 0 and lim x 0 x x 0 x does not exist. Theorem If f(x) g(x) when x is near a (except possibly at a) and the limits of f and g both exist as x approaches a, then lim f(x) lim g(x). 5

6 Theorem (The Squeeze Theorem) If f(x) g(x) h(x) when x is near a (except possibly at a) and then lim g(x) = L. Example Prove that lim x 0 x 2 cos x = 0. Example (pg 42) Show that lim θ 0 sin θ θ =. I will show that, when θ is close to zero, we have lim f(x) = lim h(x) = L, cos θ sin θ θ Then the squeeze theorem gives us our result. This is not hard if you believe that, when 0 < θ < π 2 θ < tan θ and sin θ θ <.. we have Example Show that lim x 0 sin 8x 3x. 6

7 .5 - Continuity Definition A function f is continuous at a number a if lim f(x) = f(a). Note The definition requires all of the following to be true:. f(a) is defined 2. lim exists 3. lim f(x) = f(a) Definition If f is defined near a, we say that f is discontinuous at a if f is not continuous at a. Example Determine where each of the functions is discontinuous. (a) f(x) = x2 + x 2 x (b) f(x) = { x if x 2 if x = (c) f(x) = { x 2 +x 2 x if x 0 if x = (d) f(x) = [[x]]. Note Types of discontinuity: Removable (a), Infinite Discontinuity (b), Jump Discontinuities (c,d). Example Show that f is continuous on (, ). { sin x if x < π/4 f(x) = cos x if x π/4 Definition A function f is continuous from the right at a number a if lim + f(x) = f(a) and f is continuous from the left at a if lim f(x) = f(a). Example Refer to (d) above. Definition A function f is continuous on an interval if it is continuous at every number in the interval. Theorem If f and g are continuous at a and c is a constant, then the following functions are also continuous at a: () f ± g (2) cf (3) fg f g if g(a) 0. Theorem (a) Any polynomial is continuous everywhere; that is, it is continuous on R = (, ). (b) Any rational function is continuous whenever it is defined; that is, it is continuous on its domain. Example Find lim x x 2 + 3x. 2 3x Theorem The following types of functions are continuous at every number in their domains: polynomials, rational functions, root functions, trigonometric functions. Theorem If f is continuous at b and lim g(x) = b, then lim f(g(x)) = f(b), that is, ( ) lim f(g(x)) = f lim g(x). Theorem If g is continuous at a and f is continuous at g(a), then the composite function f g given by (f g)(x) = f(g(x)) is continuous at a. 7

8 Theorem (The Intermediate Value Theorem) Suppose that f is continuous on the closed interval [a, b] and let N be any number between f(a) and f(b), where f(a) f(b). Then there exists a number c in (a, b) such that f(c) = N. (Diagram) Example Show that there is a root of the equation 5x 2 3x + 2 = 0 on the interval (0, ). 8

9 .6 Limits Involving Infinity Definition The notation lim f(x) = means that the values of f(x) can be made arbitrarily large by taking x sufficiently close to a but not equal to a. We say, the limit of f(x), as x approaches a, is infinity. Example Show that lim x 0 x = 2 Note We can define lim f(x) = similarly. Definition The line x = a is a vertical asymptote of the curve y = f(x) if at least one of the following statements is true: lim f(x) = lim f(x) = lim lim f(x) = f(x) = lim lim f(x) = + f(x) = + Example Find lim 6 x 5 x 5 and lim x x 5. Where are its vertical asymptotes? Definition Let f be a function defined on some interval (a, ). Then lim x f(x) = L means that the values of f(x) can be made as close to L as we like by taking x sufficiently large. We say the limit of f(x), as x approaches infinity, is L. Example Show that lim x x+ x =. Note We can define lim x f(x) = L similarly. Definition The line y = L is called a horizontal asymptote of the curve y = f(x) if either lim f(x) = L or lim x = L. x Note The limit laws from Section.4 continue to hold for limits at infinity. In addition, if n is a positive integer, then lim x x n = 0 Example Use the above to prove lim x x+ x =. Example Evaluate lim x x 2 +2 x 3 +x 2. lim x x n = 0. Note In general, we want to multiply the top and bottom by in the denominator. Why not the highest power in general? x n where n is the highest power that appears Example Evaluate lim x ( x2 + 3x x 2 + 5x ). Example (pg 63, example 7) Evaluate lim x sin x. Example (pg 63, example 8) Evalauate lim x sin x. Note It is possible to combine the previous two ideas to get lim x f(x) =. Example Evaluate lim x (x 2 x 4 ). Note We now provide the precise definitions of the previous limit definitions in this section. Definition Let f be a function defined on some open interval that contains the number a, except possibly at a itself. Then lim f(x) = means that for every positive number M there is a positive number δ such that if 0 < x a < δ then f(x) > M. (diagram) 9

10 Example (pg 64, example 2) Use the previous definition to prove that lim x 0 x 2 =. Definition Let f be a function defined on some interval (a, ). Then lim x f(x) = L means that for every ε > 0 there is a corresponding number N such that if x > N then f(x) L < ε. (diagram) Example (pg 65, example 3) Use the previous definition to prove that lim x x = 0. Definition Let f be a function defined on some interval (a, ). Then lim x f(x) = means that for every positive number M there is a corresponding positive number N such that if x > N then f(x) > M. Note All the previous definitions can be rewritten for instead of. 0