hw06-sol - Homework 6 Due Never 1 New Material 1 Use the...

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Unformatted text preview: Homework # 6 Due: Never 1 New Material 1. Use the guidelines of this section to sketch the curve. (a) f ( x ) = 20 x 3- 3 x 4 i. Domain f is a polynomial, so its domain is all real numbers. ii. Intercepts The y-intercept is f (0) = 0. The x-intercepts are where f ( x ) = 0, so 20 x 3- 3 x 4 = 0. Factor out an x 3 and get x 3 (20- 3 x ) = 0, so x = 0 , 20 / 3 iii. Symmetry There are no trig functions, so it’s not periodic. f (- x ) = 20(- x ) 3- 3(- x ) 4 =- 20 x 3- 3 x 4 which is neither f ( x ) nor- f ( x ), so f is neither even nor odd. iv. Asymptotes. It’s not undefined anywhere, so there won’t be any vertical asymptotes. lim x →∞ 20 x 3- 3 x 4 = lim x →∞ x 3 (20- 3 x ) =-∞ Using the same method, you can see that lim x →-∞ f ( x ) = ∞ and so there are no horizontal asymptotes. v. Intervals of Increase and Decrease f ( x ) = 60 x 2- 12 x 3 . To look for intervals of increase and decrease, we first find the critical numbers. f ( x ) is a polynomial, so it’s defined everywhere. The only critical numbers are where f ( x ) = 0. 60 x 2- 12 x 3 = 12 x 2 (5- x ), so f ( x ) = 0 when x = 0 , 5. This gives us 3 intervals: x f ( x ) x < + < x < 5 + 5 < x- Increasing: (-∞ , 0), (0 , 5) Decreasing: (5 , ∞ ) vi. Minima and Maxima At 0, the function doesn’t change direction, so it’s neither a local max nor a min. At 5, the function changes from increasing to decreasing, so it’s a maximum. Local Maximum: x = 5 Local Minimum: None vii. Intervals of concavity and inflection points. f 00 ( x ) = 120 x- 36 x 2 . Inflection points can be where this is zero or undefined. It’s never undefined, so we’re only looking for f 00 ( x ) = 0. 120 x- 36 x 2 = 0 so 12 x (10- 3 x ) = 0. Thus possible inflection points are x = 0 , 10 / 3 and we have 3 intervals to check. x f 00 ( x ) x <- < x < 10 / 3 + 10 / 3 < x- CU: (0 , 10 / 3) CD: (-∞ , 0), (10 / 3 , ∞ ) Inflection points at x = 0 , 10 / 3 since the concavity changes at both points. viii. Graph-7.5-5-2.5 2.5 5 7.5-1000-750-500-250 250 500 (b) f ( x ) = x 2 x 2- 9 i. Domain It’s a rational function, so its domain is everywhere except where the denom- inator is 0. Thus the domain is all numbers except x = 3 ,- 3. ii. Intercepts y-intercept: f (0) =- 9 = 0, so the y-intercept is 0. x-intercept: x 2 x 2- 9 = 0, so x = 0 is the x-intercept. iii. Symmetry f (- x ) = (- x ) 2 (- x ) 2- 9 = x 2 x 2- 9 = f ( x ) so the function is even. iv. Asymptotes The function is undefined at x = 3 ,- 3, but we’re only interested in the positive x values since f is even. Let’s look at one of the one-sided limits at x = 3. lim x → 3 + x 2 x 2- 9 = lim x → 3 + 1 1- 9 /x 2 = ∞ It’s ∞ since the denominator goes to zero and the numerator goes to 1, so the whole thing is an infinite limit. It’s positive because when x > 3, 1- 9 /x 2 > 0....
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hw06-sol - Homework 6 Due Never 1 New Material 1 Use the...

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