Chapter 11

The demand function for a certain make of replacement cartridges for a water purifier is given by $$ p=-0.01 x^{2}-0.1 x+6 $$ where \(p\) is the unit price in dollars and \(x\) is the quantity demanded each week, measured in units of a thousand. Determine the consumers' surplus if the market price is set at \(\$ 4 /\) cartridge.

Evaluate the definite integral. $$\int_{0}^{2} x\left(x^{2}-1\right)^{3} d x$$

Find the area of the region under the graph of the function \(f\) on the interval \([a, b]\), using the fundamental theorem of calculus. Then verify your result using geometry. $$f(x)=2 ;[1,4]$$

Find the indefinite integral. $$\int 4(4 x+3)^{4} d x$$

Verify directly that \(F\) is an antiderivative of \(f\) $$F(x)=\frac{1}{3} x^{3}+2 x^{2}-x+2 ; f(x)=x^{2}+4 x-1$$

The demand function for a certain brand of \(\mathrm{CD}\) is given by $$ p=-0.01 x^{2}-0.2 x+8 $$ where \(p\) is the wholesale unit price in dollars and \(x\) is the quantity demanded each week, measured in units of a thousand. Determine the consumers' surplus if the wholesale market price is set at \(\$ 5 /\) disc.

Evaluate the definite integral. $$\int_{0}^{1} x^{2}\left(2 x^{3}-1\right)^{4} d x$$

Find the area of the region under the graph of the function \(f\) on the interval \([a, b]\), using the fundamental theorem of calculus. Then verify your result using geometry. $$f(x)=4 ;[-1,2]$$

Find the indefinite integral. $$\int 4 x\left(2 x^{2}+1\right)^{7} d x$$

Verify directly that \(F\) is an antiderivative of \(f\) $$F(x)=x e^{x}+\pi ; f(x)=e^{x}(1+x)$$

It is known that the quantity demanded of a certain make of portable hair dryer is \(x\) hundred units/week and the corresponding wholesale unit price is $$ p=\sqrt{225-5 x} $$ dollars. Determine the consumers' surplus if the wholesale market price is set at \(\$ 10 /\) unit.

Evaluate the definite integral. $$\int_{0}^{1} x \sqrt{5 x^{2}+4} d x$$

Find the area of the region under the graph of the function \(f\) on the interval \([a, b]\), using the fundamental theorem of calculus. Then verify your result using geometry. $$f(x)=2 x ;[1,3]$$

Let \(f(x)=3 x\). a. Sketch the region \(R\) under the graph of \(f\) on the interval \([0,2]\) and find its exact area using geometry. b. Use a Riemann sum with four subintervals of equal length \((n=4)\) to approximate the area of \(R\). Choose the representative points to be the left endpoints of the subintervals. c. Repeat part (b) with eight subintervals of equal length \((n=8) .\) d. Compare the approximations obtained in parts (b) and (c) with the exact area found in part (a). Do the approximations improve with larger \(n\) ?

Find the indefinite integral. $$\int\left(x^{3}-2 x\right)^{2}\left(3 x^{2}-2\right) d x$$

Verify directly that \(F\) is an antiderivative of \(f\) $$F(x)=\sqrt{2 x^{2}-1} ; f(x)=\frac{2 x}{\sqrt{2 x^{2}-1}}$$

Evaluate the definite integral. $$\int_{1}^{3} x \sqrt{3 x^{2}-2} d x$$

Find the area of the region under the graph of the function \(f\) on the interval \([a, b]\), using the fundamental theorem of calculus. Then verify your result using geometry. $$f(x)=-\frac{1}{4} x+1 ;[1,4]$$

Find the indefinite integral. $$\int\left(3 x^{2}-2 x+1\right)\left(x^{3}-x^{2}+x\right)^{4} d x$$

Verify directly that \(F\) is an antiderivative of \(f\) $$F(x)=x \ln x-x ; f(x)=\ln x$$

Evaluate the definite integral. $$\int_{0}^{2} x^{2}\left(x^{3}+1\right)^{3 / 2} d x$$

Find the area of the region under the graph of the function \(f\) on the interval \([a, b]\). $$f(x)=2 x+3 ;[-1,2]$$

Let \(f(x)=4-2 x\) a. Sketch the region \(R\) under the graph of \(f\) on the interval \([0,2]\) and find its exact area using geometry. b. Use a Riemann sum with five subintervals of equal length \((n=5)\) to approximate the area of \(R .\) Choose the representative points to be the left endpoints of the subintervals. c. Repeat part (b) with ten subintervals of equal length \((n=10)\) d. Compare the approximations obtained in parts (b) and (c) with the exact area found in part (a). Do the approximations improve with larger \(n\) ?

Find the indefinite integral. $$\int \frac{4 x}{\left(2 x^{2}+3\right)^{3}} d x$$

(a) verify that \(G\) is an antiderivative of \(f\), (b) find all antiderivatives of \(f\), and \((c)\) sketch the graphs of a few of the family of antiderivatives found in part (b). $$G(x)=2 x, f(x)=2$$

The management of the Titan Tire Company has determined that the quantity demanded \(x\) of their Super Titan tires/week is related to the unit price \(p\) by the relation $$ p=144-x^{2} $$ where \(p\) is measured in dollars and \(x\) is measured in units of a thousand. Titan will make \(x\) units of the tires available in the market if the unit price is $$ p=48+\frac{1}{2} x^{2} $$ dollars. Determine the consumers' surplus and the producers' surplus when the market unit price is set at the equilibrium price.

Evaluate the definite integral. $$\int_{1}^{5}(2 x-1)^{5 / 2} d x$$

Find the area of the region under the graph of the function \(f\) on the interval \([a, b]\). $$f(x)=4 x-1 ;[2,4]$$

Find the indefinite integral. $$\int \frac{3 x^{2}+2}{\left(x^{3}+2 x\right)^{2}} d x$$

(a) verify that \(G\) is an antiderivative of \(f\), (b) find all antiderivatives of \(f\), and \((c)\) sketch the graphs of a few of the family of antiderivatives found in part (b). $$G(x)=2 x^{2} ; f(x)=4 x$$

The quantity demanded \(x\) (in units of a hundred) of the Mikado miniature cameras/week is related to the unit price \(p\) (in dollars) by $$ p=-0.2 x^{2}+80 $$ and the quantity \(x\) (in units of a hundred) that the supplier is willing to make available in the market is related to the unit price \(p\) (in dollars) by $$ p=0.1 x^{2}+x+40 $$ If the market price is set at the equilibrium price, find the consumers' surplus and the producers' surplus.

Evaluate the definite integral. $$\int_{0}^{1} \frac{1}{\sqrt{2 x+1}} d x$$

Find the area of the region under the graph of the function \(f\) on the interval \([a, b]\). $$f(x)=-x^{2}+4 ;[-1,2]$$

Let \(f(x)=x^{2}\) and compute the Riemann sum of \(f\) over the interval \([2,4]\), using a. Two subintervals of equal length \((n=2)\). b. Five subintervals of equal length \((n=5)\). c. Ten subintervals of equal length \((n=10)\). In each case, choose the representative points to be the midpoints of the subintervals. d. Can you guess at the area of the region under the graph of \(f\) on the interval \([2,4]\) ?

Find the indefinite integral. $$\int 3 t^{2} \sqrt{t^{3}+2} d t$$

(a) verify that \(G\) is an antiderivative of \(f\), (b) find all antiderivatives of \(f\), and \((c)\) sketch the graphs of a few of the family of antiderivatives found in part (b). $$G(x)=\frac{1}{3} x^{3} ; f(x)=x^{2}$$

Find the area of the region under the graph of the function \(f\) on the interval \([a, b]\). $$f(x)=4 x-x^{2} ;[0,4]$$

Find the indefinite integral. $$\int 3 t^{2}\left(t^{3}+2\right)^{3 / 2} d t$$

(a) verify that \(G\) is an antiderivative of \(f\), (b) find all antiderivatives of \(f\), and \((c)\) sketch the graphs of a few of the family of antiderivatives found in part (b). $$G(x)=e^{x} ; f(x)=e^{x}$$

Suppose an investment is expected to generate income at the rate of $$ R(t)=200,000 $$ dollars/year for the next 5 yr. Find the present value of this investment if the prevailing interest rate is \(8 \% /\) year compounded continuously.

Sketch the graph and find the area of the region bounded below by the graph of each function and above by the \(x\) -axis from \(x=a\) to \(x=b\). $$f(x)=-x^{2} ; a=-1, b=2$$

Evaluate the definite integral. $$\int_{1}^{2}(2 x-1)^{4} d x$$

Find the area of the region under the graph of the function \(f\) on the interval \([a, b]\). $f(x)=\frac{1}{x} ;[1,2]$$

Find the indefinite integral. $$\int\left(x^{2}-1\right)^{9} x d x$$

Find the indefinite integral. $$\int 6 d x$$

Camille purchased a \(15-\mathrm{yr}\) franchise for a computer outlet store that is expected to generate income at the rate of $$ R(t)=400,000 $$ dollars/year. If the prevailing interest rate is \(10 \% /\) year compounded continuously, find the present value of the franchise.

Sketch the graph and find the area of the region bounded below by the graph of each function and above by the \(x\) -axis from \(x=a\) to \(x=b\). $$f(x)=x^{2}-4 ; a=-2, b=2$$

Evaluate the definite integral. $$\int_{1}^{2}(2 x+4)\left(x^{2}+4 x-8\right)^{3} d x$$

Find the area of the region under the graph of the function \(f\) on the interval \([a, b]\). $$f(x)=\frac{1}{x^{2}},[2,4]$$

Let \(f(x)=x^{3}\) and compute the Riemann sum of \(f\) over the interval \([0,1]\), using a. Two subintervals of equal length \((n=2)\). b. Five subintervals of equal length \((n=5)\). c. Ten subintervals of equal length \((n=10)\). In each case, choose the representative points to be the midpoints of the subintervals. d. Can you guess at the area of the region under the graph of \(f\) on the interval \([0,1] ?\)