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Question: Find the limit. Use l’Hospital’s

Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.





Transcribed Image Text:

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> For this problem, use the following multiple regression equation: a. Interpret the meaning of the slopes. b. Interpret the meaning of the Y intercept.

> Use l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule to help find the asymptotes of f. Then use them, together with information from f' and f", to sketch the graph of f. Check your work with a graphing device. f(x) = (In x)/x

> Use l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule to help find the asymptotes of f. Then use them, together with information from f' and f", to sketch the graph of f. Check your work with a graphing device. f) — хе*

> Find the limit. Use l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule where appropriate. If there is a more elementary method, consider using it. If l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule doesn&acirc;&#128;&#153;t apply, explain why.

> Find the limit. Use l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule where appropriate. If there is a more elementary method, consider using it. If l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule doesn&acirc;&#128;&#153;t apply, explain why.

> Find the limit. Use l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule where appropriate. If there is a more elementary method, consider using it. If l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule doesn&acirc;&#128;&#153;t apply, explain why.

> Find the limit. Use l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule where appropriate. If there is a more elementary method, consider using it. If l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule doesn&acirc;&#128;&#153;t apply, explain why.

> Find the limit. Use l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule where appropriate. If there is a more elementary method, consider using it. If l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule doesn&acirc;&#128;&#153;t apply, explain why.

> Given that which of the following limits are indeterminate forms? For those that are not an indeterminate form, evaluate the limit where possible. lim f(x) = 0 lim g(x) = 0 lim h(x) = 1 %3D エ→ エ→ロ lim p(x) = 0 lim q(x) = 0 %3D (a) lim [f(x) – p(x)]

> Find the limit. Use l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule where appropriate. If there is a more elementary method, consider using it. If l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule doesn&acirc;&#128;&#153;t apply, explain why.

> Consider the following problem: A farmer with 750 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible total area of the four pens? (a). Draw

> Find the limit. Use l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule where appropriate. If there is a more elementary method, consider using it. If l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule doesn&acirc;&#128;&#153;t apply, explain why.

> Find the limit. Use l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule where appropriate. If there is a more elementary method, consider using it. If l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule doesn&acirc;&#128;&#153;t apply, explain why.

> Find the limit. Use l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule where appropriate. If there is a more elementary method, consider using it. If l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule doesn&acirc;&#128;&#153;t apply, explain why.

> Find the limit. Use l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule where appropriate. If there is a more elementary method, consider using it. If l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule doesn&acirc;&#128;&#153;t apply, explain why.

> Find the limit. Use l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule where appropriate. If there is a more elementary method, consider using it. If l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule doesn&acirc;&#128;&#153;t apply, explain why.

> Find the limit. Use l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule where appropriate. If there is a more elementary method, consider using it. If l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule doesn&acirc;&#128;&#153;t apply, explain why.

> Find the limit. Use l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule where appropriate. If there is a more elementary method, consider using it. If l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule doesn&acirc;&#128;&#153;t apply, explain why.

> Find the limit. Use l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule where appropriate. If there is a more elementary method, consider using it. If l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule doesn&acirc;&#128;&#153;t apply, explain why.

> Find the limit. Use l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule where appropriate. If there is a more elementary method, consider using it. If l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule doesn&acirc;&#128;&#153;t apply, explain why.

> Given that which of the following limits are indeterminate forms? For those that are not an indeterminate form, evaluate the limit where possible. lim f(x) = 0 lim g(x) = 0 lim h(x) = 1 %3D エ→ エ→ロ lim p(x) = 0 lim q(x) = 0 %3D (b) lim [h(x)p(x)] (a

> Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f' and f" to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. e f(x) = x2

> Find the limit. Use l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule where appropriate. If there is a more elementary method, consider using it. If l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule doesn&acirc;&#128;&#153;t apply, explain why.

> Find the limit. Use l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule where appropriate. If there is a more elementary method, consider using it. If l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule doesn&acirc;&#128;&#153;t apply, explain why.

> Find the limit. Use l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule where appropriate. If there is a more elementary method, consider using it. If l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule doesn&acirc;&#128;&#153;t apply, explain why.

> Let (a). Show that f is continuous at 0. (b). Investigate graphically whether f is differentiable at by zooming in several times toward the point (0, 1) on the graph of f. (c). Show that f is not differentiable at 0. How can you reconcile this fact wi

> If f' is continuous, use l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule to show that Explain the meaning of this equation with the aid of a diagram. f(x + h) – f(x – h) lim = f'(x) 2h

> For what values of and is the following equation true? lim x→0 (sin 2x/x3 + a + b/x2)

> Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f' and f" to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. f(x) = 4x 32

> Investigate the family of curves given by f (x) = xe-cx, where c is a real number. Start by computing the limits as x→±∞. Identify any transitional values of where the basic shape changes. What happens to the maximum or minimum points and inflection poin

> (a). Graph the function. (b). Use l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule to explain the behavior as x&acirc;&#134;&#146;0. (c). Estimate the minimum value and intervals of concavity. Then use calculus to find the exact values. f(x) —

> The figure shows a sector of a circle with central angle &Icirc;&cedil;. Let A (&Icirc;&cedil;)be the area of the segment between the chord PR and the arc PR. Let B(&Icirc;&cedil;) be the area of the triangle PQR. Find lim &Icirc;&cedil;&acirc;&#134;&#14

> Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f' and f" to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. f(x) = x² –

> (a). Investigate the family of polynomials given by the equation f (x) = 2x3 + cx2 + 2x. For what values of does the curve have maximum and minimum points? (b). Show that the minimum and maximum points of every curve in the family lie on the curve y = x

> (a). Investigate the family of polynomials given by the equation f (x) = cx4 – 2x2 + 1. For what values of does the curve have minimum points? (b). Show that the minimum and maximum points of every curve in the family lie on the parabola y = 1 – x2. Illu

> Investigate the family of curves given by the equation f (x) = x4 + cx2 + x. Start by determining the transitional value of at which the number of inflection points changes. Then graph several members of the family to see what shapes are possible. There

> Describe how the graph of f varies as varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when c changes. You should also

> If f' is the function considered in Example 3, use a computer algebra system to calculate f' and then graph it to confirm that all the maximum and minimum values are as given in the example. Calculate f" and use it to estimate the intervals of concavity

> Sketch the graph by hand using asymptotes and intercepts, but not derivatives. Then use your sketch as a guide to producing graphs (with a graphing device) that display the major features of the curve. Use these graphs to estimate the maximum and minimum

> Produce graphs of f that reveal all the important aspects of the curve. Estimate the intervals of increase and decrease and intervals of concavity, and use calculus to find these intervals exactly. 2 x 10 f(x - x*

> Two light sources of identical strength are placed 10 m apart. An object is to be placed at a point P on a line l parallel to the line joining the light sources and at a distance meter from it (see the figure). We want to locate P on l so that the intens

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> (a). Graph the function. (b). Explain the shape of the graph by computing the limit as x→0 or as x→∞. (c). Estimate the maximum and minimum values and then use calculus to find the exact values. (d). Use a graph of f" to estimate the x-coordinates of the

> (a). Graph the function. (b). Use l&acirc;&#128;&#153;Hospital&acirc;&#128;&#153;s Rule to explain the behavior as x&acirc;&#134;&#146;0. (c). Estimate the minimum value and intervals of concavity. Then use calculus to find the exact values. 1/1 f(x

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