91Ó°ÊÓ

Use a Riemann sum to approximate the area under the graph of \(f(x)\) on the given interval, with selected points as specified. \(f(x)=x^{2} ;-2 \leq x \leq 2, n=4,\) midpoints of subintervals

Find the value of \(k\) that makes the antidifferentiation formula true. [Note: You can check your answer without looking in the answer section. How?] $$\int(3 x+2)^{4} d x=k(3 x+2)^{5}+C$$

The velocity at time \(t\) seconds of a ball thrown up into the air is \(v(t)=-32 t+75\) feet per second. (a) Compute the displacement of the ball during the time interval \(1 \leq t \leq 3\) (b) Is the position of the ball at time \(t=3\) higher than its position at time \(t=1 ?\) Justify your answer. (c) Repeat part (a) using the time interval \(1 \leq t \leq 5\)

Suppose that the marginal cost function of a handbag manufacturer is \(C^{\prime \prime}(x)=\frac{3}{12} x^{2}-x+200\) dollars per unit at production level \(x\) (where \(x\) is measured in units of 100 handbags). (a) Find the total cost of producing 6 additional units if 2 units are currently being produced. (b) Describe the answer to part (a) as an area. (Give a written description rather than a sketch.)

Use a Riemann sum to approximate the area under the graph of \(f(x)\) on the given interval, with selected points as specified. \(f(x)=x^{3} ; 1 \leq x \leq 3, n=5,\) left endpoints

Find the volume of the solid of revolution generated by revolving about the \(x\)-axis the region under each of the following curves. \(y=\sqrt{x}\) from \(x=0\) to \(x=4\) (The solid generated is called a paraboloid.)

Profit Suppose that the marginal profit function for a company is \(P^{\prime}(x)=100+50 x-3 x^{2}\) at production level \(x\) (a) Find the extra profit earned from the sale of 3 additional units if 5 units are currently being produced. (b) Describe the answer to part (a) as an area. (Do not make a sketch.)

The velocity of a skydiver at time \(t\) seconds is \(v(t)=45-45 e^{-0.2 t}\) meters per second. Find the distance traveled by the skydiver the first 9 seconds.

Find the value of \(k\) that makes the antidifferentiation formula true. [Note: You can check your answer without looking in the answer section. How?] $$\int(2 x-1)^{3} d x=k(2 x-1)^{4}+C$$

Find the volume of the solid of revolution generated by revolving about the \(x\)-axis the region under each of the following curves. $$y=2 x-x^{2} \text { from } x=0 \text { to } x=2$$