91Ó°ÊÓ

Proof Let \(s(x)\) and \(c(x)\) be two functions satisfying \(s^{\prime}(x)=c(x)\) and \(c^{\prime}(x)=-s(x)\) for all \(x .\) If \(s(0)=0\) and \(c(0)=1,\) prove that \([s(x)]^{2}+[c(x)]^{2}=1\)

Choosing an Integral You are asked to find one of the integrals. Which one would you choose? Explain. (a) \(\int \sqrt{x^{3}+1} d x\) or \(\int x^{2} \sqrt{x^{3}+1} d x\) (b) \(\int \cot 2 x d x\) or \(\int \cot ^{3} 2 x \csc ^{2} 2 x d x\)

In Exercises \(81-86,\) find \(F^{\prime}(x)\) . $$F(x)=\int_{0}^{\sin x} \sqrt{t} d t$$

Depreciation The rate of depreciation \(d V / d t\) of a machine is inversely proportional to the square of \((t+1),\) where \(V\) is the value of the machine \(t\) years after it was purchased. The initial value of the machine was \(\$ 500,000,\) and its value decreased \(\$ 100,000\) in the first year. Estimate its value after 4 years.

In Exercises \(81-86,\) find \(F^{\prime}(x)\) . $$F(x)=\int_{0}^{2 x} \cos t^{4} d t$$

Water Flow Water flows from a storage tank at a rate of (500 - 5t) liters per minute. Find the amount of water that flows out of the tank during the first 18 minutes.

Oil Leak At 1:00 P.M., oil begins leaking from a tank at a rate of \((4+0.75 t)\) gallons per hour. (a) How much oil is lost from 1:00 p.m. to 4:00 p.m.? (b) How much oil is lost from 4:00 p.m. to 7:00 p.m.? (c) Compare your answers to parts (a) and (b). What do you notice?

The probability that ore samples taken from a region contain between 100\(a \%\) and 100\(b \%\) iron is $$P_{a, b}=\int_{a}^{b} \frac{1155}{32} x^{3}(1-x)^{3 / 2} d x$$ where \(x\) represents the proportion of iron. (See figure.) (a) What is the probability that a sample will contain between 0\(\%\) and 25\(\%\) iron? (b) What is the probability that a sample will contain between 50\(\%\) and 100\(\%\) iron?

In Exercises 93-98, the velocity function, in feet per second, is given for a particle moving along a straight line, where t is the time in seconds. Find (a) the displacement and (b) the total distance that the particle travels over the given interval. $$v(t)=5 t-7, \quad 0 \leq t \leq 3$$

Rate of Growth Let \(r^{\prime}(t)\) represent the rate of growth of a dog, in pounds per year. What does \(r(t)\) represent? What does \(\int_{2}^{6} r^{\prime}(t) d t\) represent about the dog?