Mathematics

1.

Evaluate the summation from k equals 0 to 4 of the sum of 3 times k and 3 . (5 points)


2.

Use geometry to evaluate the integral from negative 2 to 6 of f of x, dx where f of x equals the absolute value of x for x between negative 2 and 0 inclusive, equals x minus 2 for x greater than 0 and less than or equal to 2, and equals 3 for x greater than 2 and less than or equal to 6 . (5 points)




3.

Find F ‘(x) for F of x equals the integral from 1 to x of the square root of the quantity 1 minus t squared, dt . (5 points)



4.

Find the antiderivative of f of x equals the 5th root of x squared . (5 points)



5.

Evaluate the integral of the quotient of 1 and the square root of the quantity 1 minus 2 times x, dx . (5 points)



6.

Use your calculator to evaluate the integral from 0 to pi over 4 of the product of x and the sine of x, dx . (5 points)



7.

R is the 4th quadrant region enclosed by the x-axis and the curve y = x2 – 2kx, where k > 0. Find the value of k so that the area of the region R is 36 square units. (5 points)



8.

The base of a solid is bounded by the curve y equals the square root of the quantity x plus 2 , the x-axis and the line x = 1. The cross sections, taken perpendicular to the x-axis, are squares. Find the volume of the solid. (5 points)



9.

Find the average value of f(x) = cos(x) on the interval [0, pi over 2 ]. (5 points)



10.

Find the domain for the particular solution to the differential equation dy dx equals the quotient of negative 1 times x and y , with initial condition y(2) = 2. (5 points)



11.

The slope field for a differential equation is shown in the figure. Determine the general solution of this equation.

slope field with positive slopes in quadrants 2 and 4 and negative slopes in quadrants 1 and 4 (5 points)



12.

The function f is continuous on the interval [4, 15], with some of its values given in the table above. Estimate the average value of the function with a Trapezoidal Approximation, using the 4 intervals between those given points. (5 points)

x 4 9 11 14 15
f(x) -6 -11 -18 -21 -25


13.

If the graph of f ‘(x) has an x-intercept at x = c, which of the following must be true? (5 points)



14.

Which of the following shows that f(x) grows faster than g(x)? (5 points)



15.

Find F ‘(x) for F of x equals the integral from x cubed to 2 of the sine of t raised to the 4th power, dt . (5 points)


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