Calculate the following definite integrals
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Calculate the following definite integrals
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Given the graph of above, calculate the following definite integrals.
1. A rectangle has a vertex on the -axis between 0 and 9, a vertex at the point (9,0), a vertex on the line , and a vertex on the curve . Find the dimensions of the rectangle having the maximum possible area.
2. To get the best view of the Statue of Liberty you should be at the position where is a maximum. If the statue stands 92 meters high, including the pedestal, which is 46 meters high, how far from the base should you be?
3. You run a small furniture business. You sign a deal with a customer to deliver some number of chairs, the exact number to be determined by the customer later. The price will be $90 per chair up to 300 chairs, and above 300, the price will be reduced by $0.25 per chair (on the whole order) for every additional chair over 300 ordered. What is the maximum amount of money your company can make on this order?
4. For which positive number is largest?
Evaluate each limit.
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Velamur Quiz 8 *Note: Inflection points are not on Exam 2.
Velamur Quiz 9 *Note: Inflection points are not on Exam 2.
1. A spherical snowball is melting. Its radius is decreasing at 0.2 cm per hour when the radius is 15 cm. How fast is its volume decreasing at that time?
2. A potter forms a piece of clay into a cylinder. As he rolls it, the length, , of the cylinder increases and the radius, , decreases. If the length of the cylinder is increasing at 0.1 cm per second, find the rate at which the radius is changing when the radius is 1 cm and the length is 5 cm.
3. A certain quantity of gas occupies a volume of 20cc at a pressure of 1 atmosphere. The gas expands without the addition of heat, so, for some constant , its pressure, , and volume, , satisfy the equation . (a) Find the rate of change of pressure with respect to volume. (b) The volume is increasing at 2cc/min when the volume is 30cc. At that moment, is the pressure increasing or decreasing? How fast?
4. (a) A hemispherical bowl of radius 10cm contains water to a depth of cm. Find the radius of the surface of the water as a function of . (b) The water level drops at a rate of 0.1 cm per hour. At the what rate is the radius of the water decreasing when the depth is 5 cm?
Calculate for each of the following equations.
Find the derivative of each of the following functions.
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1. When an electric current passes through two resistors with resistance and , connected in parallel, the combined resistance, , can be calculated from the equation . Find the rate at which the combined resistance changes with respect to changes in . Assume that is constant.
2. A museum has decided to sell one of its paintings and invest the proceeds. If the picture is sold between the years 2000 and 2020 and the money from the sale is invested in a bank account earning 5% interest per year compounded annually, then , the balance in the year 2020, depends on the year, , in which the painting is sold and the sale price . If is measured from the year 2000 so that then .
a. Explain why is given by this formula.
b. Show that the formula for is equivalent to .
c. Find , given that and .
3. Let be the gas consumption (in liters/km) of a car going at velocity (in km/hr). In other words, tell you how many liters of gas the car uses to go one kilometer, if it is going at velocity . You are told that and .
a. Let be the distance the same car goes on one liter of gas at velocity . What is the relationship between and ? Find and .
b. Let be the gas consumption in liters per hour. In other words, tell you how many liters of gas the car uses in one hour if it is going at velocity . What is the relationship between and ? Find and .
c. How would you explain the practical meaning of the values of these functions and their derivatives to a driver who knows no calculus?
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