10 Multiple Choice Sigma Notation and the Summation questions

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10 Multiple Choice Sigma Notation and the Summation questions

1.

Use the properties of sigma notation and the summation formulas to evaluate the summation from i equals 1 to 10 of the quantity 2 times i squared plus 4 times i minus 5 . (5 points)


2.

Let f of x equals 4 for x less than or equal to 3 and equals the quantity 7 minus x for x greater than 3 . Use geometric formulas to evaluate the integral from x equals 1 to x equals 7 of f of x, dx . (5 points)



3.

Write the definite integral for the summation: the limit as n goes to infinity of the summation from k equals 1 to n of the product of the square of the quantity 1 plus k over n squared and 1 over n . (5 points)



4.

Find the derivative with respect to x of the integral from 1 to x cubed of the quantity e raised to the 2t power, dt . (5 points)



5.

Find an antiderivative of x squared divided by 3 plus C . (5 points)



6.

Evaluate the integral of the quotient of cosine x and cosine cubed of x, dx . (5 points)



7.

Evaluate the integral the integral of the product of 4 times x and the 5th power of the quantity x squared minus 3, dx . (5 points)



8.

Find the antiderivative of the product of 6 times x and e raised to the x squared power . (5 points)



9.

Use your calculator to evaluate the definite integral from 1 to 2 of the natural log of the absolute value of the quantity x squared plus 1, dx . Give 3 decimal places for your answer. (5 points)




10.

Suppose the integral from 2 to 10 of g of x, dx equals 10 and the integral from 8 to 10 of g of x, dx equals negative 6' , find the value of the integral from 2 to 8 of one-half times g of x, dx . (5 points)

1.

Using n = 4 equal-width rectangles, approximate the integral from negative 2 to 2 of the quantity x cubed plus 8, dx . Use the mid-point of each sub-interval to determine the height of each rectangle. (10 points)

2.

Water leaks from a tank at the rate of r(t) gallons per hour. The rate decreased as time passed, and values of the rate at two-hour time intervals are shown in the table below. The total amount of water that leaked out is evaluated by a Riemann sum. Find the upper estimate (left end-points of each rectangle) for the total amount of water that leaked out by using five rectangles.

Give your answer with one decimal place. (10 points)

t (hr) 0 2 4 6 8 10
r(t) (gal/hr) 10.7 8.6 6.6 5.2 5.0 4.5


3.

Find the interval on which the curve of y equals the integral from 0 to x of 2 divided by the quantity 1 plus 3 times t plus t squared, dt is concave up. (10 points)


4.

Evaluate the integral of the quotient of the sine of x and the square root of the quantity 1 plus cosine x, dx . (10 points)


5.

Evaluate exactly the value of the integral from negative 1 to 0 of the product of the cube of quantity 2 times x to the 5th power plus 6 times x and 5 times x to the 4th power plus 3, dx . Your work must include the use of substitution and the antiderivative. (10 points)


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