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angements of the letters of DEADWOOD be arranged? the word DEADWOOD are there if the arrangement must begin and end with the letter D? 1014 CHAPTER 11 seQuences, proBaBility and counting theory BInOMIAl THeOReM 23 ξ€ͺ. 43. Evaluate the binomial coefficient ξ€’ _ 8 1 y ξ€ͺ 44. Use the Binomial Theorem to expand ξ€’ 3x + _ 2 6. ...
of the sequence defined by the explicit formula an = n2 βˆ’ n βˆ’ 1 ________. n! 4. An arithmetic sequence has the first term a1 = βˆ’4 4 _ and common difference d = βˆ’. What is the 6th term? 3 6. Write an explicit formula for the arithmetic sequence 15.6, 15, 14.4, 13.8, … and then find the 32nd term. 1 1 _ _, βˆ’ 7. Is the s...
create the β€œTop-Four” list out of the 32 employees? 19. A rock group needs to choose 3 songs to play at the annual Battle of the Bands. How many ways can they choose their set if have 15 songs to pick from? 20. A self-serve frozen yogurt shop has 8 candy toppings and 4 fruit toppings to choose from. How many ways are ...
his top speed at every instant. How then, do we approximate his speed at any given instant? We will find the answer to this and many related questions in this chapter. 1017 1018 CHAPTER 12 introduction to calculus leARnInG OBjeCTIVeS In this section, you will: β€’ Understand limit notation. β€’ Find a limit using a graph....
indicates that as x approaches a both from the left of x = a and the right of x = a, the output value approaches L. Consider the function We can factor the function as shown. f (x) = x 2 βˆ’ 6x βˆ’ 7 __________ x βˆ’ 7. f (x) =  (x βˆ’ 7) (x + 1) ____________  x βˆ’ 7 f (x) = x + 1, x β‰  7 Cancel like factors in numerator and ...
a), the corresponding output values of f (x) get closer to L. Note that the value of the limit is not affected by the output value of f (x) at a. Both a and L must be real numbers. We write it as lim x β†’ a f (x) = L Example 1 Understanding the Limit of a Function For the following limit, define a, f (x), and L. lim x ...
The input values that approach 7 from the left in Figure 3 are 6.9, 6.99, and 6.999. The corresponding outputs are 7.9, 7.99, and 7.999. These values are getting closer to 8. The limit of values of f (x) as x approaches from the left is known as the left-hand limit. For this function, 8 is the left-hand limit of the f...
approaches 7 from the right x Figure 4 The left- and right-hand limits are the same for this function. SECTION 12.1 Finding limits: numerical and graphical approaches 1021 left- and right-hand limits The left-hand limit of a function f (x) as x approaches a from the left is equal to L, denoted by x β†’ aβˆ’ f(x) = L. lim ...
a x y = f(x) a x < a x > a x Figure 5 To determine if a left-hand limit exists, we observe the branch of the graph to the left of x = a, but near x = a. This is where x < a. We see that the outputs are getting close to some real number L so there is a left-hand limit. To determine if a right-hand limit exists, observe...
β†’ 2+ y iii. lim x β†’ 2 f (x) iv. f (2) 10 3βˆ’4 βˆ’1 βˆ’2 βˆ’1 βˆ’2 f 21 3 4 5 6 7 8 x f y 10 3βˆ’4βˆ’5βˆ’6 βˆ’1 βˆ’2 βˆ’1 βˆ’2 21 3 4 5 6 x Solution a. Looking at Figure 6: Figure 6 Figure 7 i. ii. iii. iv. lim x β†’ 2βˆ’ f (x) = 8; when x < 2, but infinitesimally close to 2, the output values get close to y = 8. lim x β†’ 2+ f (x) = 3; when x > 2...
4 3 2 1 βˆ’3βˆ’4βˆ’5βˆ’6 βˆ’1 βˆ’2 βˆ’1 βˆ’2 βˆ’3 f 21 3 4 5 6 x Figure 8 SECTION 12.1 Finding limits: numerical and graphical approaches 1023 Finding a limit Using a Table Creating a table is a way to determine limits using numeric information. We create a table of values in which the input values of x approach a from both sides. Then...
if it exists, by evaluating the function at values near x = 0. We cannot find a function value for x = 0 directly because the result would have a denominator equal to 0, and thus would be undefined. lim x β†’ 0 f (x) = 5sin(x) _ 3x We create Figure 10 by choosing several input values close to x = 0, with half of them le...
utilities aren’t available, and they can be calculated to a higher precision than could be seen with an unaided eye inspecting a graph. Example 4 Using a Graphing Utility to Determine a Limit With the use of a graphing utility, if possible, determine the left- and right-hand limits of the following function as x appro...
APTER 12 introduction to calculus 12.1 SeCTIOn exeRCISeS VeRBAl 1. Explain the difference between a value at x = a and the limit as x approaches a. 2. Explain why we say a function does not have a limit as x approaches a if, as x approaches a, the left-hand limit is not equal to the right-hand limit. GRAPHICAl For the ...
not exist. f (x) = 5, lim x β†’ 5 f (x) = 0, f (5) = 4, f (x) = 2, lim 20. lim x β†’ βˆ’3 x β†’ 1+ f (βˆ’3) = 0, f (0) = 0 f (x) = βˆ’2, lim x β†’ 3 f (x) = βˆ’ 4, 21. lim x β†’ Ο€ f (Ο€) = √ f (x) = Ο€ 2, lim Ο€ _ f (x) =, lim 2 x β†’ 1βˆ’ 2, f (0) does not exist. x β†’ βˆ’Ο€ β€” f (x) = 0, For the following exercises, use a graphing calculator to d...
x 2 βˆ’ 6x βˆ’ 7 _ x 2 βˆ’ 7x ; a = 7 34. f (x) = 1 βˆ’ x 2 _ x 2 βˆ’ 3x + 2 ; a = 1 31. f (x 33. f (x) = x 2 βˆ’ 1 _ x 2 βˆ’ 3x + 2 ; a = 1 35. f (x) = 10 βˆ’ 10x2 _ x 2 βˆ’ 3x + 2 ; a = 1 36. f (x) = x __ 6x 2 βˆ’ 5x βˆ’ 6 3 __ ; a = 2 37. f (x) = x __ 4x 2 + 4x + 1 1 __ ; a = βˆ’ 2 38. f (x For the following exercises, use a calculator to...
and c is the speed of light. Find the limit of the mass, m, as v approaches c βˆ’. 52. Allow the speed of light, c, to be equal to 1.0. If the mass, m, is 1, what occurs to m as v β†’ c? Using the values listed in Table 1, make a conjecture as to what the mass is as v approaches 1.00. v 0.5 0.9 0.95 0.99 0.999 0.99999 Tab...
a. Look again at Figure 1 and Figure 2. Notice that in both graphs, as x approaches 7, the output values approach 8. This means Remember that when determining a limit, the concern is what occurs near x = a, not at x = a. In this section, we will use a variety of methods, such as rewriting functions by factoring, to ev...
x β†’ a [f (x) β‹… g(x)] = lim x β†’ a f (x) β‹… lim x β†’ a g(x) = A β‹… B lim x β†’ a f (x) ____ g(x) = lim f (x) A x β†’ a _________ _ = g(x) lim B x β†’ a, B β‰  0 lim x β†’ a [ f (x)]n = ξ€° lim x β†’ ∞ f (x) ξ€² n = An, where n is a positive integer n lim x β†’ a √ β€” n √ f (x) = β€” lim x β†’ a [ f (x)] = n √ β€” A lim x β†’ a p(x) = p(a) Table 1 Ex...
β†’ 3 (x2) (5x2) = 5 lim x β†’ 3 = 5(32) = 45 Constant times a function property Function raised to an exponent property Example 3 Evaluating the Limit of a Polynomial Algebraically Evaluate lim x β†’ 5 (2x3 βˆ’ 3x + 1). Solution lim x β†’ 5 (2x3 βˆ’ 3x + 1) = lim x β†’ 5 = 2 lim x β†’ 5 (2x3) βˆ’ lim x β†’ 5 (3x) + lim x β†’ 5 (x) + lim x...
often need to rewrite the function algebraically before applying the properties of a limit. If the denominator evaluates to 0 when we apply the properties of a limit directly, we must rewrite the quotient in a different form. One approach is to write the quotient in factored form and simplify. How To… Given the limit ...
ξ€ͺ 5x ξ€’ 1 1 __ __ ξ€ͺ βˆ’ x 5 __________ 5x(x βˆ’ 5) ξ€’ Multiply numerator and denominator by LCD. = lim x β†’ 5 1 1 ξ€ͺ ξ€ͺ βˆ’ 5x ξ€’ 5x ξ€’ __ __ 5 x _________________ 5x(x βˆ’ 5) ξ€ͺ ξ€’ ξ€’ ξ€’ = lim x β†’ 5 5 βˆ’ x ________ 5x(x βˆ’ 5) = lim x β†’ 5 βˆ’1(x βˆ’ 5) _________ 5x(x βˆ’ 5) ξ€ͺ ξ€ͺ 1 _ 5x βˆ’ = lim x β†’ 5 1 ____ 5(5) = βˆ’ Apply distributive property. S...
0 β€” β€” ξ€ͺ = lim ξ€’ x β†’ 0 25 βˆ’ x + 5 25 βˆ’ x βˆ’ 5 √ √ __ __ β‹… x 25 βˆ’ x + 5 √ ξ€ͺ β€” = lim x β†’ 0 (25 βˆ’ x) βˆ’ 25 ξ€ͺ ξ€’ __ 25 βˆ’ x + 5 ξ€ͺ x ξ€’ √ β€” __ = lim ξ€’ x β†’ 0 25 βˆ’ x __ = lim ξ€ͺ ξ€’ x β†’ 0 25 βˆ’ x + 5 ξ€ͺ x ξ€’ √ β€” βˆ’x = = β€” βˆ’1 __ 25 βˆ’ 0 + 5 √ βˆ’1 _____ 5 + 5 1 ___ 10 = βˆ’ Multiply numerator and denominator by the conjugate. Multiply: ξ€’ √ β€” 2...
found, choose several values close to and on either side of the input where the function is undefined. 3. Use the numeric evidence to estimate the limits on both sides. Example 9 Evaluating the Limit of a Quotient with Absolute Values Evaluate lim Solution The function is undefined at x = 7, so we will try values clos...
+ 18x βˆ’ 35 __ 2x + 7 ξ€ͺ 11. lim x2 βˆ’ 9 ξ€ͺ ξ€’ _ x 2 βˆ’ 5x + 6 x β†’ 3 14. lim h β†’ 0 ξ€ͺ ξ€’ (3 + h)3 βˆ’ 27 __ __ x 18. lim x β†’ 0 15. lim h β†’ 0 19. lim x β†’ 9 12. lim x β†’ βˆ’3 x β†’ 3 βˆ’12x4 + 108x2 ξ€ͺ 13. lim ξ€’ ξ€ͺ ξ€’ x2 + 2x βˆ’ 3 βˆ’7x4 βˆ’ 21x3 _ __ x βˆ’ 3 ξ€’ ξ€ͺ (h + 3)2 βˆ’ 9 ξ€ͺ ξ€’ 5 βˆ’ h βˆ’ √ √ __ __ h h ξ€’ ξ€ͺ ξ€’ x __ β€” 1 + 2x βˆ’ 1 √ x βˆ’ x2 √ _ β€” 1 βˆ’ √ ...
+ 2x + 1, x ≀ 0 x βˆ’ 3, x > 0 ; lim x β†’ 0+ f (x) 38. f (x) =  2x2 + 2x + 1, x ≀ 0 x βˆ’ 3, x > 0 ; lim x β†’ 0βˆ’ f (x) 39. f (x) =  2x2 + 2x + 1, x ≀ 0 x βˆ’ 3, x > 0 ; lim x β†’ 0 f (x) 40. lim x β†’ 4 β€” x + 5 βˆ’ 3 √ ___________ x βˆ’ 4 41. lim x β†’ 2+ ξ€’ 2x βˆ’ γ€šxγ€› ξ€ͺ 42. lim x β†’ 2 β€” x + 7 βˆ’ 3 √ ___________ x2 βˆ’ x βˆ’ 2 43. lim x β†’ 3+ ...
1βˆ’1 βˆ’2 βˆ’3 βˆ’4 βˆ’5 21 3 4 5 x Figure 5 Figure 6 ReAl-WORlD APPlICATIOnS 58. The position function s(t) = βˆ’16t 2 + 144t gives the position of a projectile as a function of time. Find the average velocity (average rate of change) on the interval [1, 2]. 60. The amount of money in an account after t years compounded continuo...
, and by 4 p.m. it was 118Β° F. Sometime between 2 a.m. and 4 p.m., the temperature outside must have been exactly 110.5Β° F. In fact, any temperature between 96Β° F and 118Β° F occurred at some point that day. This means all real numbers in the output between 96Β° F and 118Β° F are generated at some point by the function ac...
) represented in Figure 3 as an example. y f a Figure 3 x Condition 1 According to Condition 1, the function f (a) defined at x = a must exist. In other words, there is a y-coordinate at x = a as in Figure 4. y f(a) f a Figure 4 x f (x), must exist. This means that at x = a the Condition 2 According to Condition 2, at ...
a left-hand limit and a right-hand limit even if they are not equal. If the left- and right-hand limits exist but are different, the graph β€œjumps” at x = a. The function is said to have a jump discontinuity. As an example, look at the graph of the function y = f (x) in Figure 10. Notice as x approaches a how the outpu...
x approaches 5 is 8, and Condition 2 is not satisfied. This means there is a removable discontinuity at x = 5. b. Condition 2 is satisfied because g(2) = βˆ’ 2. Notice that the function is a piecewise function, and for each piece, the function is defined everywhere on its domain. Let’s examine Condition 1 by determining...
Ex: f (x) = 2ln(x), x > 0 Tangent functions Rational functions Ο€ __ + kΟ€, k is an integer Ex: f (x) = tan(x) + 2, x β‰  2 Ex: f (x) = x 2 βˆ’ 25 _______ x βˆ’ 7, x β‰  7 Table 2 How To… Given a function f (x), determine if the function is continuous at x = a. 1. Check Condition 1: f (a) exists. 2. Check Condition 2: lim x β†’ a...
οΏ½ Condition 2 fails. 8 __ b. x = 3 8 ξ€ͺ exist? Condition 1: Does f ξ€’ _ 3 Condition 2: Does lim f (x) exist? x β†’ 8 _ 3 32 β‡’ Condition 1 is satisfied. 8 8 _ _ To the left of x = f (x) = 4x; to the right of x = 3 3 8 _. right-hand limits as x approaches 3 8 _ ξ€ͺ = 4 ξ€’ f (x) = lim 3 β€’ Left-hand limit: lim 32 x) = 8 + x. We n...
satisfied. At x = 5, there exists a removable discontinuity. See Figure 12. f y 13 12 11 10 21 3 4 5 6 7 8 Figure 12 x Try It #3 Determine whether the function f (x) = 9 βˆ’ x2 ______ x2 βˆ’ 3x is continuous at x = 3. If not, state the type of discontinuity. Determining the Input Values for Which a Function Is Discontinuo...
f (x) = 3 on 2 ≀ x < 4, and f (x) = x2 βˆ’ 5 on x β‰₯ 4. Polynomial functions are continuous everywhere. Any discontinuities would be at the boundary points, x = 2 and x = 4. At x = 2, let us check the three conditions of continuity. Condition 1: f (2) = 3 β‡’ Condition 1 is satisfied. Condition 2: Because a different funct...
#4 Determine where the function f (x) = x < 2 Ο€x _, 4 Ο€ _ x, 2Ο€x, x > 6 2 ≀ x ≀ 6 is discontinuous. Determining Whether a Function Is Continuous To determine whether a piecewise function is continuous or discontinuous, in addition to checking the boundary points, we must also check whether each of the functions that m...
) 1048 CHAPTER 12 introduction to calculus 12.3 SeCTIOn exeRCISeS VeRBAl 1. State in your own words what it means for a 2. State in your own words what it means for a function f to be continuous at x = c. function to be continuous on the interval (a, b). AlGeBRAIC For the following exercises, determine why the function...
| 18. f (x) = 25 βˆ’ x2 __ x2 βˆ’ 10x + 25, a = 5 20. f (x) = x3 βˆ’ 27 _ x2 βˆ’ 3x, a = 3 22. f (x2 For the following exercises, determine whether or not the given function f is continuous everywhere. If it is continuous everywhere it is defined, state for what range it is continuous. If it is discontinuous, state where it i...
what x-coordinates is the function discontinuous? 45. What condition of continuity is violated at these points? x 5 10 x Figure 15 y 10 5 0 βˆ’5 βˆ’10 Figure 16 βˆ’10 βˆ’5 y 5 4 3 2 1 βˆ’5 βˆ’4 βˆ’3 βˆ’1 βˆ’2 βˆ’1 βˆ’2 βˆ’3 βˆ’4 βˆ’5 βˆ’6 21 3 4 5 x Figure 17 1050 CHAPTER 12 introduction to calculus 46. Consider the function shown in Figure 18. At...
times per day. Supposedly, this average is up from 10 years ago when the average teenager opened a refrigerator door 20 times per day.[37] It is estimated that a television is on in a home 6.75 hours per day, whereas parents spend an estimated 5.5 minutes per day having a meaningful conversation with their children. T...
f (a + h)) for some positive-value h. 37 http://www.csun.edu/science/health/docs/tv&health.html Source provided. 1052 CHAPTER 12 introduction to calculus y f(a + h) (a, f(a + h)) f f(a + h) f(a) (a, f(a )) h a a + h x Figure 2 Connecting point a with a point just beyond allows us to measure a slope close to that of a ...
f (βˆ’1) = 5. AROC = f (a + h) βˆ’ f (a) _____________ h 5 βˆ’ (βˆ’6) ________ βˆ’3 = = 11 ___ βˆ’3 = βˆ’ 11 ___ 3 SECTION 12.4 derivatives 1053 Try It #1 Find the average rate of change connecting the points (βˆ’5, 1.5) and ( βˆ’ 2.5, 9). Understanding the Instantaneous Rate of Change Now that we can find the average rate of change, s...
a, is given by f β€²(a) = lim h β†’ 0 f (a + h) βˆ’ f (a) _____________ h The expression is called the difference quotient. f (a + h) βˆ’ f (a) _____________ h We use the difference quotient to evaluate the limit of the rate of change of the function as h approaches 0. Derivatives: Interpretations and Notation The derivative ...
f (a) _____________ h. Example 2 Finding the Derivative of a Polynomial Function Find the derivative of the function f (x) = x 2 βˆ’ 3x + 5 at x = a. Solution We have: f β€²(a) = lim h β†’ 0 f (a + h) βˆ’ f (a) _____________ h Substitute f (a + h) = (a + h)2 βˆ’ 3(a + h) + 5 and f (a) = a2 βˆ’ 3a + 5. Definition of a derivative (...
ξ€ͺ ξ€² _______________________________________ (2 βˆ’ (a + h))(2 βˆ’ a)(h) = lim h β†’ 0 Substitute f (a + h) and f (a). Multiply numerator and denominator by (2 βˆ’ (a + h))(2 βˆ’ a). (2 βˆ’ (a + h))(2 βˆ’ a) ξ€’ 3 + (a + h) ___________ (2 βˆ’ (a + h)) (2 βˆ’ (a + h))(2 βˆ’ a)(h) ξ€ͺ βˆ’ (2 βˆ’ (a + h))  (2 βˆ’ a) ξ€’ 3 + a ξ€ͺ _____  2 βˆ’ a ___________...
+ h + 4 √ 4 √ __ 1056 CHAPTER 12 introduction to calculus β€” β€” a + h βˆ’ 4 √ 4 √ ξ€’ ______________ f β€²(a) = lim √ __ ξ€ͺ 16(a + h) βˆ’ 16a ξ€’ __ = lim h β†’ 0 h4 ξ€’ √ a + h + 4 √ β€” β€” Multiply. ξ€ͺ β‹… ξ€’ ξ€ͺ a ξ€ͺ 16a + 16h βˆ’   16a ξ€ͺ ξ€’ __ = lim β€” h β†’ 0 a ξ€ͺ h4 ξ€’ √ a + h + 4 √ β€” Distribute and combine like terms. Simplify. Evaluate the li...
of distance per time, or velocity. Notice that the initial height is 6 feet. To find the instantaneous velocity, we find the derivative and evaluate it at t = 1 and t = 3: f β€²(a) = lim h β†’ 0 f (a + h) βˆ’ f (a) _____________ h = lim h β†’ 0 βˆ’16(t + h)2 + 64(t + h) + 6 βˆ’ (βˆ’16t 2 + 64t + 6) _________________________________...
in that section of the curve. Extend the line far enough to calculate its slope as change in y __________. change in x Example 6 Estimating the Derivative at a Point on the Graph of a Function From the graph of the function y = f (x) presented in Figure 5, estimate each of the following: f (0); f (2); f β€²(0); f β€²(2) 1...
1βˆ’1 βˆ’2 βˆ’3 βˆ’4 βˆ’5 βˆ’5 βˆ’4 βˆ’3 βˆ’2 21 3 4 5 x Figure 7 SECTION 12.4 derivatives 1059 Using Instantaneous Rates of Change to Solve Real-World Problems Another way to interpret an instantaneous rate of change at x = a is to observe the function in a real-world context. The unit for the derivative of a function f (x) is output u...
+ h2 βˆ’ 100h ______________ h = h(2x + h βˆ’ 100) ______________ h = 2x + h βˆ’ 100 = 2x βˆ’ 100 f β€²(x) = 2x βˆ’ 100 f β€²(200) = 2(200) βˆ’ 100 = 300 Formula for a derivative Substitute f (a + h) and f (a). Multiply polynomials, distribute. Collect like terms. Factor and cancel like terms. Simplify. Evaluate when h = 0. Formula f...
’s Derivative Does Not Exist To understand where a function’s derivative does not exist, we need to recall what normally happens when a function f (x) has a derivative at x = a. Suppose we use a graphing utility to zoom in on x = a. If the function f (x) is differentiable, that is, if it is a function that can be diffe...
1βˆ’2βˆ’3βˆ’4βˆ’5 βˆ’2 1 2 3 4 5 x Figure 11 The graph of f (x) has a discontinuity at x = 2. In Figure 12, we see the graph of f (x) = | x |. We see that the graph has a corner point at x = 0. y 5 4 3 2 1 βˆ’1βˆ’1 βˆ’2 βˆ’3βˆ’4βˆ’5 βˆ’2 1 2 3 4 5 x Figure 12 The graph of f (x) = |x| has a corner point at x = 0. In Figure 13, we see that the ...
At the points where the graph is discontinuous or not differentiable, state why. y 5 4 3 2 1 βˆ’1βˆ’1 βˆ’2 βˆ’3 βˆ’4 βˆ’5 βˆ’5 βˆ’4 βˆ’3 βˆ’2 1 2 3 4 5 x f Figure 15 Solution The graph of f (x) is continuous on (βˆ’βˆž, βˆ’2) βˆͺ (βˆ’2, 1) βˆͺ (1, ∞). The graph of f (x) has a removable discontinuity at x = βˆ’2 and a jump discontinuity at x = 1. See F...
the curve at x = a and is therefore equal to f β€²(a), the derivative of f (x) at x = a. The coordinate pair of the point on the line at x = a is (a, f (a)). If we substitute into the point-slope form, we have m = f β€²(a) y1 = f (a) x1 = a y = m(x βˆ’ x1) + y1 ↑ f (a) ↑ f β€²(a) ↑ a The equation of the tangent line is y = f ...
2  βˆ’4a βˆ’ 4h  βˆ’ a2  + 4a ____________________________ h = lim h β†’ 0 = lim h β†’ 0 = lim h β†’ 0 = lim h β†’ 0 2ah + h2 βˆ’ 4h ____________ h  h (2a + h βˆ’ 4) ____________  h = 2a + 0 βˆ’ 4 f β€²(a) = 2a βˆ’ 4 f β€²(3) = 2(3) βˆ’ 4 = 2 Equation of tangent line at x = 3: y = f β€²(a)(x βˆ’ a) + f (a) y = f β€²(3)(x βˆ’ 3) + f (3) y = 2(x βˆ’ 3) ...
, it would travel the specified distance. instantaneous velocity Let the function s(t) represent the position of an object at time t. The instantaneous velocity or velocity of the object at time t = a is given by sβ€²(a) = lim h β†’ 0 s(a + h) βˆ’ s(a) _____________ h Example 11 Finding the Instantaneous Velocity A ball is t...
a rate of 28 ft/sec. 1066 CHAPTER 12 introduction to calculus Try It #10 A fireworks rocket is shot upward out of a pit 12 ft below the ground at a velocity of 60 ft/sec. Its height in feet after t seconds is given by s = βˆ’16t2 + 60t βˆ’ 12. What is its instantaneous velocity after 4 seconds? Access these online resourc...
5 13. f (x) = 5 βˆ’ 2x ______ 3 + 2x 17. f (x) = 5Ο€ For the following exercises, find the average rate of change between the two points. 19. (4, βˆ’3) and (βˆ’2, βˆ’1) 18. (βˆ’2, 0) and (βˆ’4, 5) 20. (0, 5) and (6, 5) 21. (7, βˆ’2) and (7, 10) For the following polynomial functions, find the derivatives. 22. f (x) = x3 + 1 23. f (x...
4 βˆ’5 βˆ’6 βˆ’7 βˆ’8 βˆ’9 βˆ’10 34. f (βˆ’1) 40. f β€²(0) 35. f (0) 41. f β€²(1) 36. f (1) 42. f β€²(2) Figure 20 37. f (2) 43. f β€²(3) 44. Sketch the function based on the information below: f β€²(x) = 2x, f (2) = 4 TeCHnOlOGY 38. f (3) 39. f β€²(βˆ’1) 45. Numerically evaluate the derivative. Explore the behavior of the graph of f (x) = x 2 ar...
r changes 57. Find the instantaneous rate of change of V when from 1 cm to 2 cm. r = 3 cm. For the following exercises, the revenue generated by selling x items is given by R(x) = 2x 2 + 10x. 58. Find the average change of the revenue function as 59. Find Rβ€²(10) and interpret. x changes from x = 10 to x = 20. 60. Find...
x = a instantaneous rate of change the slope of a function at a given point; at x = a it is given by f β€²(a) = lim h β†’ 0 f (a + h) βˆ’ f (a) _____________ h instantaneous velocity the change in speed or direction at a given instant; a function s(t) represents the position of an object at time t, and the instantaneous vel...
intersects a curve at a single point two-sided limit the limit of a function f (x), as x approaches a, is equal to L, that is, lim x β†’ a f (x) = L if and only if x β†’ aβˆ’ f (x) = lim lim x β†’ a+ f (x). Key equations average rate of change AROC = f (a + h) βˆ’ f (a) _____________ h derivative of a function f β€²(a) = lim h β†’ ...
2 and Example 3. β€’ The limit of a function that has been raised to a power equals the same power of the limit of the function. Another method is direct substitution. See Example 4. β€’ The limit of the root of a function equals the corresponding root of the limit of the function. β€’ One way to find the limit of a functio...
Example 4. β€’ The difference quotient is the quotient in the formula for the instantaneous rate of change: β€’ Instantaneous rates of change can be used to find solutions to many real-world problems. See Example 5. f (a + h) βˆ’ f (a) __ h β€’ The instantaneous rate of change can be found by observing the slope of a function...
there is no limit. 8. f (x) =  1 _____, x + 1 (x + 1)2, if x = βˆ’2 a = βˆ’2 if x β‰  βˆ’2 7. f (x) =  9. f (x) =  | x | βˆ’ 1, x3, if x β‰  1 if, if x < 1 if x > 1 a = 1 FInDInG lIMITS: PROPeRTIeS OF lIMITS For the following exercises, find the limits if lim x β†’ c f (x) = βˆ’3 and lim x β†’ c g(x) = 5. 10. lim x β†’ c ( f (x) + g(x...
6x 2 + 23x + 20 _____________ 4x 2 βˆ’ 25 5 ___ ; a = βˆ’ 2 25. f (x) = β€” x βˆ’ 3 √ _______ 9 βˆ’ x ; a = 9 For the following exercises, determine where the given function f (x) is continuous. Where it is not continuous, state which conditions fail, and classify any discontinuities. 26. f (x) = x 2 βˆ’ 2x βˆ’ 15 30. f (x) = 1 __ ...
TeST For the following exercises, use the graph of f in Figure 1. y 5 4 3 2 1 21 3 4 5 x βˆ’1βˆ’2βˆ’3βˆ’4βˆ’5 βˆ’1 βˆ’2 βˆ’3 βˆ’4 βˆ’5 Figure 1 1. f (1) 2. lim x β†’ βˆ’1+ f (x) 3. lim x β†’ βˆ’1βˆ’ f (x) 4. lim x β†’ βˆ’1 f (x) 5. lim x β†’ βˆ’2 f (x) 6. At what values of x is f discontinuous? What property of continuity is violated? For the following ex...
βˆ’3βˆ’4βˆ’5 –1 –2 –3 –4 –5 Figure 2 1076 CHAPTER 12 introduction to calculus For the following exercises, with the aid of a graphing utility, explain why the function is not differentiable everywhere on its domain. Specify the points where the function is not differentiable. 18. f (x) = |x βˆ’ 2| βˆ’ |x + 2| 19. f (x) = 2 _____...
_ 5 ξ€ͺ. 34. Explore the behavior of the graph of f (x) around x = 1 by graphing the function on the following domains, [0.9, 1.1], [0.99, 1.01], [0.999, 1.001], and [0.9999, 1.0001]. Use this information to determine whether the function appears to be differentiable at x = 1. For the following exercises, find the deriv...
–4 –3 –2 f (x) 4 3 2 1 –1 –1 –2 –3 –4 Domain: (βˆ’βˆž, ∞) Range: (βˆ’βˆž, ∞) Figure A2 Exponential f (x) 4 3 2 1 –1 –1 –2 –3 –4 Domain: (βˆ’βˆž, 0) βˆͺ (0, ∞) Range: (βˆ’βˆž, 0) βˆͺ (0, ∞) Natural Logarithm f (x) 4 3 2 1 –1 –1 –2 –3 –4 21 3 4 x y = e x y = ln(x) 21 3 4 x –4 –3 –2 21 3 4 x –4 –3 –2 Domain: (βˆ’βˆž, ∞) Range: [0, ∞) Domain: (βˆ’...
20 15 10 5 Ο€ –5 2 –10 –15 –20 Ο€ 2 Ο€ 3Ο€ 2 2Ο€ x Domain: x β‰  Ο€k where k is an integer Range: (βˆ’βˆž, βˆ’1] βˆͺ [1, ∞) Ο€ Domain: x β‰  k where k is an odd integer 2 Range: (βˆ’βˆž, βˆ’1] βˆͺ [1, ∞) Domain: x β‰  Ο€k where k is an integer Range: (βˆ’βˆž, ∞) Inverse Sine y y = sinβˆ’1 x Figure A5 Inverse Cosine y y = cosβˆ’1 x Inverse Tangent y y = ta...
= βˆ’tan ΞΈ csc(βˆ’ΞΈ) = βˆ’csc ΞΈ cot(βˆ’ΞΈ) = βˆ’cot ΞΈ Cofunction identities Fundamental Identities Sum and Difference Identities Double-Angle Formulas Ο€ _ sin ΞΈ = cos ξ€’ βˆ’ ΞΈ ξ€ͺ 2 Ο€ _ cos ΞΈ = sin ξ€’ βˆ’ ΞΈ ξ€ͺ 2 Ο€ _ tan ΞΈ = cot ξ€’ βˆ’ ΞΈ ξ€ͺ 2 Ο€ _ cot ΞΈ = tan ξ€’ βˆ’ ΞΈ ξ€ͺ 2 Ο€ _ sec ΞΈ = csc ξ€’ βˆ’ ΞΈ ξ€ͺ 2 Ο€ _ csc ΞΈ = sec ξ€’ βˆ’ ΞΈ ξ€ͺ 2 sin ΞΈ _ cos ΞΈ 1 _ cos ΞΈ...
βˆ’ cos(2ΞΈ) _________ 2 cos2ΞΈ = 1 + cos(2ΞΈ) _________ 2 tan2ΞΈ = 1 βˆ’ cos(2ΞΈ) _________ 1 + cos(2ΞΈ) 1 __ ξ€° cos ξ€’ Ξ± βˆ’ Ξ² ξ€ͺ + cos ξ€’ Ξ± + Ξ² ξ€ͺ ξ€² cos Ξ± cos Ξ² = 2 1 __ ξ€° sin ξ€’ Ξ± + Ξ² ξ€ͺ + sin ξ€’ Ξ± βˆ’ Ξ² ξ€ͺ ξ€² sin Ξ± cos Ξ² = 2 1 __ ξ€° cos ξ€’ Ξ± βˆ’ Ξ² ξ€ͺ βˆ’ cos ξ€’ Ξ± + Ξ² ξ€ͺ ξ€² sin Ξ± sin Ξ² = 2 1 __ ξ€° sin ξ€’ Ξ± + Ξ² ξ€ͺ βˆ’ sin ξ€’ Ξ± βˆ’ Ξ² ξ€ͺ ξ€² cos Ξ± sin Ξ² = 2 sin...
, because each bank account has a single balance at any given time. b. No, because several bank account numbers may have the same balance. c. No, because the same output may 10. a. Yes, letter grade correspond to more than one input. is a function of percent grade. b. No, it is not one-to-one. There are 100 different p...
= f (x) βˆ’ g(x) = (x βˆ’ 1) βˆ’ (x2 βˆ’ 1) = x βˆ’ x 2 b. No, the functions are not the same. 2. A gravitational force is still a force, so a(G(r)) makes sense as the acceleration of a planet at a distance r from the Sun (due to gravity), but G(a(F)) does not make sense. 3. f ( g (1)) = f (3) = 3 and g ( f (4)) = g (1) = 3 4. ...
21 3 4 5 x g(x) = βˆ’f (x) = βˆ’x2 9. 8. even 10. g(x) = 3x βˆ’ 2 2 6 4 9 12 15 x 8 g(x) 0 1 __ x ξ€ͺ so using the square root function we get 11. g(x) = f ξ€’ ___ 3 1 g(x) = √ __ x 3 B-1 B-2 Section 1.6 4. x = βˆ’1 or x = 2 1. | x βˆ’ 2 | ≀ 3 2. Using the variable p for passing, | p βˆ’ 80 | ≀ 20 3. f (x) = βˆ’| x + 2 | + 3 5. f (0) =...
0 = βˆ’3(x βˆ’ 0); y = βˆ’3x 6. H(x) = 0.5x + 12.5 5. y = βˆ’7x + 3 Section 2.2 1. y 2. Possible answers include (βˆ’3, 7), (βˆ’6, 9), or (βˆ’9, 11) (0, 6) (4, 3) (8, 0) 8 10 x 42 6 –10 –8 –6 –4 3. –10 –8 y y = 2x + 4 y = x 4. (16, 0) 5. a. f (x) = 2x; b. g(x) = βˆ’ 1 __ x 2 6. y = βˆ’ 1 __ x + 6 3 42 6 8 10 x 10 8 6 4 2 –2 –2 –4 –6 –8...
(βˆ’7.5) β‰ˆ 1.64; he doesn’t make it. 2. g(x) = x 2 βˆ’ 6x + 13 in general form; g(x) = (x βˆ’ 3)2 + 4 3. The domain is all real numbers. The in standard form 8 __, or ξ€° range is f (x) β‰₯ 11 5. 3 seconds; 256 feet; 7 seconds 4. y-intercept at (0, 13), No x-intercepts, ∞ ξ€ͺ. 8 __ 11 3. The degree is 6. The leading term is βˆ’x 6. ...
a zero of –5 with multiplicity 3, a zero of –1 with multiplicity 2, and a zero of 3 with multiplicity 4. TRY IT ANSWERS B-3 y 6 5 4 3 2 1 –2 –1 –2 –3 –4 –5 –10 –8 –6 –4 642 8 10 x 1 _ Horizontal asymptote at y =. 2 Vertical asymptotes at x = 1 4 and x = 3. y-intercept at ξ€’ 0, _ ξ€ͺ. 3 x-intercepts at (2, 0) and (–2, 0)....
128 ____ 3 9 __ 2. 2 Chapter 4 Section 4.1 β€” 4. (0, 129) and (2, 236); N(t) = 129(1.3526)t 3. About 1.548 billion people; by the year 2031, 1. g(x) = 0.875x and j(x) = 1095.6βˆ’2x represent exponential functions. 2. 5.5556 India's population will exceed China's by about 0.001 billion, or 1 million people. β€” 2 )x ; Answe...
βˆ’2, and βˆ’4. 1 __ 3. There are no rational zeros. 4. The zeros are βˆ’4,, and 1. 2 5. f (x) = βˆ’ 1 x 3 + 5 __ __ 2 2 0 positive real roots and 0 negative real roots. The graph shows that there are 2 positive real zeros and 0 negative real zeros. 7. 3 meters by 4 meters by 7 meters 6. There must be 4, 2, or x 2 βˆ’ 2x + 10 S...
We then set the numerator equal to 0 and find the x-intercepts are at (2.5, 0) and (3.5, 0). Finally, we evaluate the function at 0 and find the y-intercept to be at ξ€’ 0, βˆ’35 ξ€ͺ. ____ 9 TRY IT ANSWERS B-4 4. f(x) 1 f(x) = (4)x 2 (1, 2) The domain is ( βˆ’βˆž, ∞); the range is (0, ∞); the horizontal asymptote is y = 0. (0, ...
11) 2 β€” Section 4.4 1. (2, ∞) 3. 2. (5, ∞) f(x) 4 3 2 1 –2–1 –2 –3 –10 –8 –6 –4 4. x = βˆ’4 (βˆ’1, 1) y 5 4 3 2 1 –6 –5 –4 –3 –2 –1 (βˆ’3, 0) –1 –2 –3 –4 –5, 11 5 (1, 0) 42 6 8 f (x) = log (x) x = 0 10 1 5 x = 0 f (x) = log3(x + 4) y = log3(x) 4 5 6 (3, 1) x 321 (1, 0) The domain is (0, ∞), the range is (βˆ’βˆž, ∞), and the ver...
6 8 10 Section 4.5 3. 2ln(x) 6. 2log(x) + 3log(y) βˆ’ 4log(z) 1. logb(2) + logb(2) + logb(2) + logb(k) = 3logb(2) + logb(k) 2. log3(x + 3) βˆ’ log3(x βˆ’ 1) βˆ’ log3(x βˆ’ 2) 4. βˆ’2ln(x) 5. log3(16) 2 1 __ __ ln(x βˆ’ 1) + ln(2x + 1) βˆ’ ln(x + 3) βˆ’ ln(x βˆ’ 3) 7. ln(x) 8. 2 3 3 β‹… 5 5 ____ __ 9. log ξ€’ ξ€ͺ ; can also be written log ξ€’ ξ€ͺ b...
800,000 Γ— years β‰ˆ 226,572, 993 years. Section 4.7 1. f (t) = A0 e βˆ’0.0000000087t 2. Less than 230 years; 229.3157 to be exact 3. f (t) = A0 e ξ€’ ln(2) 5. 895 cases on day 15 6. Exponential. y = 2e 0.5x 4. 6.026 hours 7. y = 3e (ln 0.5)x ____ 3 ξ€ͺ t TRY IT ANSWERS B-5 Section 4.8 1. a. The exponential regression model tha...
= 2 4 Ο€ β€” csc ξ€’ __ 2 ξ€ͺ = √ 4 β€” tan(t) = 33 __ 56 cot(t) = 56 __ 33 Ο€ tan ξ€’ __ ξ€ͺ = 1 4 Ο€ cot ξ€’ __ ξ€ͺ = 1 4 Ο€ β€” __. 3 ; missing angle is 6 4. 2 5. Adjacent = 10; opposite = 10 √ 6. About 52 ft. Chapter 6 Section 6.1 Ο€ __ 1 __ 1. 6Ο€ 3. 2. compressed ; right 2 2 1 __ 5. Midline: y = 0; Amplitude : |A| = 4. 2 units up ; 2 2...
_ = 2 sec 3 3. sin ξ€’ βˆ’ β€” β€” β€” 2 2 7Ο€ ___ ξ€ͺ = √ ____ 2 4 7Ο€ ___ β€” sec ξ€’ βˆ’ ξ€ͺ = √ 4 3 4. βˆ’ √ 5. βˆ’2 7. cot t = βˆ’ 8 __ 17 17 __ 15 8. sin t = βˆ’1 csc t = sec t = Undefined β€” 2 9. sec t = √ cot t = 1 10. β‰ˆ βˆ’2.414 –1 y 2 1 Ο€ g(x) = –2 cos 3 –7 –5 –3 –1 1 3 5 7 9. 10. tan t = βˆ’1 cot t = βˆ’1 Ο€ __ = √ tan 3 Ο€ __ = cot 3 tan ξ€’ βˆ’ β€” ...
–7 –5 –1 1 5 7 x 3. g(x) = 4tan(2x) x = βˆ’9 x = βˆ’3 x = 3 x = 9 –5 4. y 6 4 2 f (x) = βˆ’2.5sec (0.4x) This is a vertical reflection of the preceding graph because A is negative. –π Ο€ 2Ο€ 3Ο€ 4Ο€ x –2 –4 –6 5. y f(x) = βˆ’6sec (4x + 24 –8 –12 –16 –20 –24 y 6. f (x) = 0.5csc (2x) 6 4 2 –2 –4 –6 7. Ο€ 4 3Ο€ (x) = 2 sec Ο€ x 2 + 1 f...
β€” β€” β€” 6 6 3. 2 – √ √ _________ 4 2 + √ √ _________ 4 2. 1. 5. tan(Ο€ βˆ’ ΞΈ) = tan(Ο€) βˆ’ tan ΞΈ ____________ 1 + tan(Ο€)tan ΞΈ = 0 βˆ’ tan ΞΈ __________ 1 + 0 βˆ™ tan ΞΈ β€” 1 βˆ’ √ _______ 1 + √ 3 4. cos ξ€’ 3 β€” 5Ο€ ___ ξ€ͺ 14 = βˆ’tan ΞΈ Section 7.3 1. cos(2Ξ±) = 7 __ 32 2. cos4 ΞΈ βˆ’ sin4 ΞΈ = (cos2 ΞΈ + sin2 ΞΈ)(cos2 ΞΈ βˆ’ sin2 ΞΈ) = cos(2ΞΈ) 3. cos...
β€” βˆ’2 βˆ’ √ ________ 4 3 4. 2 sin(2ΞΈ)cos(ΞΈ) TRY IT ANSWERS 5. tan ΞΈ cot ΞΈ βˆ’ cos2 ΞΈ = ξ€’ cos ΞΈ _ sin ΞΈ ξ€ͺ βˆ’ cos2 ΞΈ ξ€ͺ ξ€’ sin ΞΈ _ cos ΞΈ = 1βˆ’ cos2 ΞΈ = sin2 ΞΈ Section 7.5 1. x = 7Ο€ _, 6 11Ο€ _ 6 Ο€ _ 3. ΞΈ β‰ˆ 1.7722 Β± 2Ο€k and Β± Ο€k 2. 3 3Ο€ Ο€ _ _ ΞΈ β‰ˆ 4.5110 Β± 2Ο€k 4. cos ΞΈ = βˆ’1, ΞΈ = Ο€ 5., 2 2 2Ο€ _, 3 4Ο€ _, 3 Section 7.6 2 _ 1. The ampl...
. a β‰ˆ 14.9, Ξ² β‰ˆ 23.8Β°, Ξ³ β‰ˆ 126.2Β° 2. Ξ± β‰ˆ 27.7Β°, Ξ² β‰ˆ 40.5Β°, Ξ³ β‰ˆ 111.8Β° 3. Area = 552 square feet 4. About 8.15 square feet β€” 3. (x, y) = ξ€’ √ 3 1 ξ€ͺ ____ _, βˆ’ 2 2 in the standard form for a circle, x2 + (y βˆ’ 1)2 = 1 4. r = √ β€” 3 5. x2 + y2 = 2y or, B-7 Section 8.4 1. The equation fails the symmetry test with respect to th...
2 3 _ + i 2 8. z0 = 2(cos(30Β°) + isin(30Β°)), z1 = 2(cos(120Β°) + isin(120Β°)) z2 = 2(cos(210Β°) + isin(210Β°)), z3 = 2(cos(300Β°) + isin(300Β°)) Section 8.6 1. t x(t) βˆ’1 0 1 2 βˆ’4 βˆ’3 βˆ’2 βˆ’1 y (βˆ’4, 2) (0, 2) (4, 2) (0, 0) x 2 = 8y x y = βˆ’2 y(t) 2 4 6 8 Section 8.3 1. (βˆ’15, 0) (βˆ’12, 0) y (0, 9) (0, βˆ’9) 2. y (15, 0) x (12, 0) (3...
Β°)i + √ β€” 34 sin(59Β°)j x β€” Magnitude = √ 34 5 ΞΈ = tanβˆ’1 ξ€’ _ ξ€ͺ = 59.04Β° 3 Section 9.1 1. Not a solution 2. The solution to the system is the ordered pair(βˆ’5, 3). y (βˆ’4, 0) –4 –5 –3 –2 7 6 5 4 3 2 1 –1–1 –2 –3 –4 –5 –6 –7 (0, 7) (0, 0) 2 1 (4, 0) 3 4 5 βˆ’ (0, βˆ’7) 3. (βˆ’2, βˆ’5) 4. (βˆ’6, βˆ’2) 5. (10, βˆ’4) 6. No solution. It is a...
y + 3z = 1 y + z = βˆ’9 3. (2, 1) 5. (1, 1, 1) Section 9.6  βˆ’3 1. ξ€° 4 11 4 2 3 ξ€² 4. ξ€° 5 __ 2 βˆ’ 5 __ 2 17 __  Section 9.7 1. AB = ξ€° 1 4 ξ€² ξ€° βˆ’3 βˆ’1 βˆ’3 1 6. $150,000 at 7%, $750,000 at 8%, $600,000 at 10% βˆ’4 1 1(βˆ’3) + 4(1) βˆ’1(βˆ’3) + βˆ’3(13(1) + βˆ’4(βˆ’1) ξ€² = ξ€° 1(1) + 1(βˆ’1) = ξ€° 1 0 ξ€² 1 0 3. Aβˆ’1 = ξ€° 1 2 4 βˆ’3 βˆ’5 3 ξ€² 1 2 6 1(βˆ’4) +...
, Β±7); Co-vertices: (Β±4, 0) Foci: (0, Β± √ 33 ) β€” (βˆ’4, 0) –4 –5 –3 –2 y (0, 7) (0, 0) 1 2 (4, 0) 3 4 5 βˆ’ (0, βˆ’7) 7 6 5 4 3 2 1 –1–1 –2 –3 –4 –5 –6 –7 TRY IT ANSWERS B-9 (y + 1)2 = βˆ’4(x βˆ’ 8) y y = βˆ’1 (8, βˆ’1) x = 7 y x (9, 1) (9, βˆ’1) (9, βˆ’3) y = 8 5. Center: (4, 2); Vertices: (βˆ’2, 2) and (10, 2); Co-vertices: ξ€’ 4, 2 βˆ’ 2 √...
ξ€’ Β± √ β€” 34, 0 ξ€ͺ 3. (y βˆ’ 3)2 _ 25 + (x βˆ’ 1)2 _ 144 = 1 4. Vertices: (Β±12, 0); Co-vertices: (0, Β±9); Foci: (Β±15, 0); 3 _ Asymptotes: y = Β± x 4 (βˆ’15, 0) (βˆ’12, 0) Axis of symmetry: x = βˆ’2; Focus: (βˆ’2, βˆ’2); Directrix: y = 8; Endpoints of the latus rectum: (βˆ’12, βˆ’2) and (8, βˆ’2). (βˆ’2, 3) (x + 2)2 = βˆ’20(y βˆ’ 3) (βˆ’2, βˆ’2) (βˆ’12, ...
_ 400 βˆ’ y2 _ 3600 = 1 or x2 _ 202 βˆ’ y2 _ 602 = 1. Section 10.3 1. Focus: (βˆ’4, 0); Directrix: x = 4; Endpoints of the latus rectum: (βˆ’4, Β±8) y (βˆ’4, 8) y2 = βˆ’16x (βˆ’4, 0) (0, 0) 2. Focus: (0, 2); Directrix: y = βˆ’2; Endpoints of the latus rectum: (Β±4, 2) (βˆ’4, βˆ’8) x = 4 y (βˆ’4, 2) (0, 2) (4, 2) (0, 0) 2 = 8y x y = βˆ’2 x x 2....
βˆ’15 _ (5a + 4) 2 6. βˆ’2, 0, 0, βˆ’3 2. f '(a) = 6a + 7 d. After 20 seconds, she is moving 3 _ 4. 2 7. a. After zero seconds, she b. After 10 seconds, she has traveled c. After 10 seconds, she is moving eastward 5. 0 has traveled 0 feet. 150 feet east. at a rate of 15 ft/sec. westward at a rate of 10 ft/sec. 100 feet west...
7) = 5,040 5. 12 9. 64 sundaes 10. 840 3. 120 7. P(7, 5) = 2,520 Section 11.6 1. a. 35 b. 330 2. a. x 5 βˆ’ 5x 4y + 10x 3y2 βˆ’ 10x2y3 + 5xy 4 βˆ’ y 5 b. 8x 3 + 60x 2y + 150xy 2 + 125y 3 3. βˆ’10,206x 4y 5 7 _ 13 3. 2 _ 2. 3 5 _ 5. 6. a. 6 1 _ 91 4. 2 _ 13 5 _ 91 b. c. 86 _ 91 Section 11.7 1. Outcome Probability Roll of 1 Rol...
output if the relation is to be a function. horizontal line intersects the graph of a function more than once, that indicates that for that output there is more than one input. A function is one-to-one if each output corresponds to only one input. 7. Function 13. Function 15. Function 21. Function 27. f (βˆ’3) = βˆ’11, f ...