Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error.

√146

Use the guidelines of this section to make a complete graph of f.

f(x) = 2 - 2x^2/3 + x^4/3

Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error.

ln 1.05

Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error.

e⁰·⁰⁶

First Derivative Test

a. Locate the critical points of f.

b. Use the First Derivative Test to locate the local maximum and minimum values.

c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist).

f(x) = x² + 3 on [-3,2]

Suppose you own a tour bus and you book groups of 20 to 70 people for a day tour. The cost per person is $30 minus $0.25 for every ticket sold. If gas and other miscellaneous costs are $200, how many tickets should you sell to maximize your profit? Treat the number of tickets as a nonnegative real number.

First Derivative Test

a. Locate the critical points of f.

b. Use the First Derivative Test to locate the local maximum and minimum values.

c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist).

f(x) = x√(4 - x²) on [-2,2]

Find the height h, radius r, and volume of a right circular cylinder with maximum volume that is inscribed in a sphere of radius R.

Travel costs A simple model for travel costs involves the cost of gasoline and the cost of a driver. Specifically, assume gasoline costs $p/gallon and the vehicle gets g miles per gallon. Also assume the driver earns $w/hour.

b. At what speed does the gas mileage function have its maximum?

Travel costs A simple model for travel costs involves the cost of gasoline and the cost of a driver. Specifically, assume gasoline costs $p/gallon and the vehicle gets g miles per gallon. Also assume the driver earns $w/hour.

e. Should the optimal speed be increased or decreased (compared with part (d)) if L is increased from 400 mi to 500 mi? Explain.

First Derivative Test

a. Locate the critical points of f.

b. Use the First Derivative Test to locate the local maximum and minimum values.

c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist).

f(x) = 2x⁵ - 5x⁴ - 10x³ + 4 on [-2,4]

First Derivative Test

a. Locate the critical points of f.

b. Use the First Derivative Test to locate the local maximum and minimum values.

c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist).

f(x) = x²/(x² - 1) on [-4,4]

Mean Value Theorem The population of a culture of cells grows according to the function P(t) = 100t / t+1, where t ≥ 0 is measured in weeks.

a. What is the average rate of change in the population over the interval [0, 8]?

Mean Value Theorem The population of a culture of cells grows according to the function P(t) = 100t / t+1, where t ≥ 0 is measured in weeks.

b. At what point of the interval [0, 8] is the instantaneous rate of change equal to the average rate of change?

First Derivative Test

a. Locate the critical points of f.

b. Use the First Derivative Test to locate the local maximum and minimum values.

c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist).

f(x) = x - 2 tan⁻¹ x on [-√3,√3)

Growth rate of bamboo Bamboo belongs to the grass family and is one of the fastest growing plants in the world.

a. A bamboo shoot was 500 cm tall at 10:00 A.M. and 515 cm tall at 3:00 P.M. Compute the average growth rate of the bamboo shoot in cm/hr over the period of time from 10:00 A.M. to 3:00 P.M.

Growth rate of bamboo Bamboo belongs to the grass family and is one of the fastest growing plants in the world.

b. Based on the Mean Value Theorem, what can you conclude about the instantaneous growth rate of bamboo measured in millimeters per second between 10:00 A.M. and 3:00 P.M.?

{Use of Tech} Approximating reciprocals To approximate the reciprocal of a number a without using division, we can apply Newton’s method to the function f(x) = 1/x - a.​​ 

b. Apply Newton’s method with a = 7 using a starting value of your choice. Compute an approximation with eight digits of accuracy. What number does Newton’s method approximate in this case?  

Verify that the following functions satisfy the conditions of Theorem 4.9 on their domains. Then find the location and value of the absolute extrema guaranteed by the theorem.

f(x) = 4x + 1/√x

{Use of Tech} Modified Newton’s method The function f has a root of multiplicity 2 at r if f(r) = f'(r) = 0 and f"(r) ≠ 0. In this case, a slight modification of Newton’s method, known as the modified (or accelerated) Newton’s method, is given by the formula xₙ + 1 = xₙ - (2f(xₙ)/(f'(xₙ), for n = 0, 1, 2, . . . . This modified form generally increases the rate of convergence.

b. Apply Newton’s method and the modified Newton’s method using x₀ = 0.1 to find the value of x₃ in each case. Compare the accuracy of these values of x₃.

{Use of Tech} Flow from a tank A cylindrical tank ​​is full at time t=0 when a valve in the bottom of the tank is opened. By Torricelli’s law, the volume of water in the tank after t hours is V=100(200−t)², measured in cubic meters.

c. Find the rate at which water flows from the tank and plot the flow rate function. 

Evaluate the following limits in two different ways: with and without l’Hôpital’s Rule.

lim_x→∞ (2x⁵ - x + 1) / (5x⁶ + x)

M​​inimizing related functions Complete each of the following parts.

b. What value of x minimizes ƒ(x) = (x- a₁)² + (x - a₂)² + (x - a₃)² , for constants a₁, a₂, and a₃?

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→1⁻ (x/(x-1) - x/(ln x) 

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→0⁺ (cot x - 1/x) 

The figure shows six containers, each of which is filled from the top. Assume water is poured into the containers at a constant rate and each container is filled in 10 s. Assume also that the horizontal cross sections of the containers are always circles. Let h (t) be the depth of water in the container at time t, for 0 ≤ t ≤ 10 . <IMAGE>

d. For each container, where does h' (the derivative of h ) have an absolute maximum on [0 , 10]?

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_Θ→π/2 (tan Θ - secΘ)

Local max/min of x¹⸍ˣ Use analytical methods to find all local extrema of the function ƒ(x) = x¹⸍ˣ , for x > 0 . Verify your work using a graphing utility.

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→∞ (x - √(x²+4x))

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→∞ x² ln( cos 1/x)