91Ó°ÊÓ

Calculate the left Riemann sums for the given functions over the given interval, using the given values of \(n .\) (When rounding, round answers to four decimal places.) HINT [See Example 3.] $$ f(x)=1-3 x \text { over }[-1,1], n=4 $$

The way Professor Costenoble drives, he burns gas at the rate of \(1 /(t+1)\) gallons each hour, \(t\) hours after a fill-up. Find the number of gallons of gas he burns in the first 10 hours after a fill-up.

Sales Annual sales of fountain pens in Littleville are 4,000 per year and are increasing by \(10 \%\) per year. How many fountain pens will be sold over the next five years?

The total cost of producing \(x\) items is given by $$ C(x)=246.76+\int_{0}^{x} 5 t d t $$ Find the fixed cost and the marginal cost of producing the 10th item.

Give a formula for the right Riemann Sum with \(n\) equal subdivisions \(a=x_{0}

If \(F(x)\) and \(G(x)\) are both antiderivatives of \(f(x)\), how are \(F(x)\) and \(G(x)\) related?

The work done in accelerating an object from velocity \(v_{0}\) to velocity \(v_{1}\) is given by $$ W=\int_{v_{0}}^{v_{1}} v \frac{d p}{d v} d v $$ where \(p\) is its momentum, given by \(p=m v(m=\) mass \()\). Assuming that \(m\) is a constant, show that $$ W=\frac{1}{2} m v_{1}^{2}-\frac{1}{2} m v_{0}^{2} $$ The quantity \(\frac{1}{2} m v^{2}\) is referred to as the kinetic energy of the object, so the work required to accelerate an object is given by its change in kinetic energy.

According to the special theory of relativity, the apparent mass of an object depends on its velocity according to the formula $$ m=\frac{m_{0}}{\left(1-\frac{v^{2}}{c^{2}}\right)^{1 / 2}} $$ where \(v\) is its velocity, \(m_{0}\) is the "rest mass" of the object (that is, its mass when \(v=0\) ), and \(c\) is the velocity of light: approximately \(3 \times 10^{8}\) meters per second. a. Show that, if \(p=m v\) is the momentum, $$ \frac{d p}{d v}=\frac{m_{0}}{\left(1-\frac{v^{2}}{c^{2}}\right)^{3 / 2}} $$ b. Use the integral formula for \(W\) in the preceding exercise, together with the result in part (a) to show that the work required to accelerate an object from a velocity of \(v_{0}\) to \(v_{1}\) is given by $$ W=\frac{m_{0} c^{2}}{\sqrt{1-\frac{v_{1}^{2}}{c^{2}}}}-\frac{m_{0} c^{2}}{\sqrt{1-\frac{v_{0}^{2}}{c^{2}}}} . $$ We call the quantity \(\frac{m_{0} c^{2}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\) the total relativistic energy of an object moving at velocity \(v\). Thus, the work to accelerate an object from one velocity to another is given by the change in its total relativistic energy. c. Deduce (as Albert Einstein did) that the total relativistic energy \(E\) of a body at rest with rest mass \(m\) is given by the famous equation $$ E=m c^{2} $$.

At what stage of a calculation using a \(u\) substitution should you substitute back for \(u\) in terms of \(x\) : before or after taking the antiderivative?

\- Give an example of a nonzero function whose definite integral over the interval \([4,6]\) is zero.