Chapter 4

A car comes to a stop six seconds after the driver applies the brakes. While the brakes are on, the following velocities are recorded: $$ \begin{array}{|l|l|l|l|l|} \hline \text { Time since brakes applied (sec) } & 0 & 2 & 4 & 6 \\ \hline \text { Velocity (ft/s) } & 88 & 45 & 16 & 0 \\ \hline \end{array} $$ Give lower and upper estimates (using all of the available data) for the distance the car traveled after the brakes were applied. lower: upper: (for each, include units) On a sketch of velocity against time, show the lower and upper estimates you found above..

Your task is to estimate how far an object traveled during the time interval \(0 \leq t \leq 8\), but you only have the following data about the velocity of the object. $$ \begin{array}{|l|l|l|l|l|l|l|l|l|l|} \hline \text { time (sec) } & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline \text { velocity (feet/sec) } & -3 & -2 & -3 & -1 & -2 & -1 & 4 & 1 & 2 \\\ \hline \end{array} $$ To get an idea of what the velocity function might look like, you pick up a black pen, plot the data points, and connect them by curves. Your sketch looks something like the black curve in the graph below. You decide to use a left endpoint Riemann sum to estimate the total displacement. So, you pick up a blue pen and draw rectangles whose height is determined by the velocity measurement at the left endpoint of each one-second interval. By using the left endpoint Riemann sum as an approximation, you are assuming that the actual velocity is approximately constant on each one-second interval (or, equivalently, that the actual acceleration is approximately zero on each one- second interval), and that the velocity and acceleration have discontinuous jumps every second. This assumption is probably incorrect because it is likely that the velocity and acceleration change continuously over time. However, you decide to use this approximation anyway since it seems like a reasonable approximation to the actual velocity given the limited amount of data. (A) Using the left endpoint Riemann sum, find approximately how far the object traveled. Your answers must include the correct units. Total displacement \(=\) ___________________ Total distance traveled \(=\) _________________ Using the same data, you also decide to estimate how far the object traveled using a right endpoint Riemann sum. So, you sketch the curve again with a black pen, and draw rectangles whose height is determined by the velocity measurement at the right endpoint of each one-second interval. (B) Using the right endpoint Riemann sum, find approximately how far the object traveled. Your answers must include the correct units. Total displacement \(=\) ________________ Total distance traveled \(=\) ______________

Find the average value of \(f(x)=4 x+4\) over [4,9] average value \(=\) _________________

Find the average value of \(f(x)=6 x+5\) over [2,6] average value = _____________

On a sketch of \(y=e^{x},\) represent the left Riemann sum with \(n=2\) approximating \(\int_{2}^{3} e^{x} d x\). Write out the terms of the sum, but do not evaluate it: Sum \(=\) ___________________ + ___________________ On another sketch, represent the right Riemann sum with \(n=2\) approximating \(\int_{2}^{3} e^{x} d x\). Write out the terms of the sum, but do not evaluate it: Sum \(=\) ___________________ + ___________________ Which sum is an overestimate? Which sum is an underestimate?

Let \(S\) be the sum given by $$ S=\left((1.4)^{2}+1\right) \cdot 0.4+\left((1.8)^{2}+1\right) \cdot 0.4+\left((2.2)^{2}+1\right) \cdot 0.4+\left((2.6)^{2}+1\right) \cdot 0.4+\left((3.0)^{2}+1\right) \cdot 0.4 $$ a. Assume that \(S\) is a right Riemann sum. For what function \(f\) and what interval \([a, b]\) is \(S\) an approximation of the area under \(f\) and above the \(x\) -axis on \([a, b] ?\) Why? b. How does your answer to (a) change if \(S\) is a left Riemann sum? a middle Riemann sum? c. Suppose that \(S\) really is a right Riemann sum. What is geometric quantity does \(S\) approximate? d. Use sigma notation to write a new sum \(R\) that is the right Riemann sum for the same function, but that uses twice as many subintervals as \(S .\)

Suppose that an accelerating car goes from 0 mph to 66.8 mph in five seconds. Its velocity is given in the following table, converted from miles per hour to feet per second, so that all time measurements are in seconds. (Note: \(1 \mathrm{mph}\) is \(22 / 15\) feet per sec \(=22 / 15 \mathrm{ft} / \mathrm{s}\).) Find the average acceleration of the car over each of the first two seconds. $$ \begin{array}{|l|l|l|l|l|l|l|} \hline t & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline v(t) & 0.00 & 33.41 & 57.91 & 75.73 & 89.09 & 98.00 \\ \hline \end{array} $$ average acceleration over the first second \(=\) ______________ (include units) average aceleration over the second second = _______________ (include units)

The velocity function is \(v(t)=t^{2}-6 t+8\) for a particle moving along a line. Find the displacement (net distance covered) of the particle during the time interval [-2,5] displacement \(=\) ______________

A function \(f\) is given piecewise by the formula $$ f(x)=\left\\{\begin{array}{ll} -x^{2}+2 x+1, & \text { if } 0 \leq x<2 \\ -x+3, & \text { if } 2 \leq x<3 \\ x^{2}-8 x+15, & \text { if } 3 \leq x \leq 5 \end{array}\right. $$ a. Determine the exact value of the net signed area enclosed by \(f\) and the \(x\) -axis on the interval [2,5] b. Compute the exact average value of \(f\) on [0,5] . c. Find a formula for a function \(g\) on \(5 \leq x \leq 7\) so that if we extend the above definition of \(f\) so that \(f(x)=g(x)\) if \(5 \leq x \leq 7,\) it follows that \(\int_{0}^{7} f(x) d x=0\)

Suppose that the velocity of a moving object is given by \(v(t)=t(t-1)(t-3),\) measured in feet per second, and that this function is valid for \(0 \leq t \leq 4\) a. Write an expression involving definite integrals whose value is the total change in position of the object on the interval [0,4] . b. Use appropriate technology (such as http://gvsu.edu/s/a93) to compute Riemann sums to estimate the object's total change in position on [0,4] . Work to ensure that your estimate is accurate to two decimal places, and explain how you know this to be the case. c. Write an expression involving definite integrals whose value is the total distance traveled by the object on [0,4] d. Use appropriate technology to compute Riemann sums to estimate the object's total distance travelled on [0,4] . Work to ensure that your estimate is accurate to two decimal places, and explain how you know this to be the case. e. What is the object's average velocity on [0,4] , accurate to two decimal places?

A toy rocket is launched vertically from the ground on a day with no wind. The rocket's vertical velocity at time \(t\) (in seconds) is given by \(v(t)=500-32 t\) feet \(/ \mathrm{sec}\). a. At what time after the rocket is launched does the rocket's velocity equal zero? Call this time value \(a\). What happens to the rocket at \(t=a\) ? b. Find the value of the total area enclosed by \(y=v(t)\) and the \(t\) -axis on the interval \(0 \leq t \leq a\). What does this area represent in terms of the physical setting of the problem? c. Find an antiderivative \(s\) of the function \(v\). That is, find a function \(s\) such that \(s^{\prime}(t)=v(t)\). d. Compute the value of \(s(a)-s(0) .\) What does this number represent in terms of the physical setting of the problem? e. Compute \(s(5)-s(1)\). What does this number tell you about the rocket's flight?

When an aircraft attempts to climb as rapidly as possible, its climb rate (in feet per minute) decreases as altitude increases, because the air is less dense at higher altitudes. Given below is a table showing performance data for a certain single engine aircraft, giving its climb rate at various altitudes, where \(c(h)\) denotes the climb rate of the airplane at an altitude \(h\). $$ \begin{array}{lllllllllll} \hline h \text { (feet) } & 0 & 1000 & 2000 & 3000 & 4000 & 5000 & 6000 & 7000 & 8000 & 9000 & 10,000 \\ \hline c(\mathrm{ft} / \mathrm{min}) & 925 & 875 & 830 & 780 & 730 & 685 & 635 & 585 & 535 & 490 & 440 \\ \hline \end{array} $$ Let a new function called \(m(h)\) measure the number of minutes required for a plane at altitude \(h\) to climb the next foot of altitude. a. Determine a similar table of values for \(m(h)\) and explain how it is related to the table above. Be sure to explain the units. b. Give a careful interpretation of a function whose derivative is \(m(h)\). Describe what the input is and what the output is. Also, explain in plain English what the function tells us. c. Determine a definite integral whose value tells us exactly the number of minutes required for the airplane to ascend to 10,000 feet of altitude. Clearly explain why the value of this integral has the required meaning. d. Use the Riemann sum \(M_{5}\) to estimate the value of the integral you found in (c). Include units on your result.

In Chapter \(1,\) we showed that for an object moving along a straight line with position function \(s(t),\) the object's "average velocity on the interval \([a, b]^{\prime \prime}\) is given by $$ A V_{[a, b]}=\frac{s(b)-s(a)}{b-a} $$ More recently in Chapter \(4,\) we found that for an object moving along a straight line with velocity function \(v(t),\) the object's "average value of its velocity function on \([a, b]^{\prime \prime}\) is $$ v_{\mathrm{AVG}[a, b]}=\frac{1}{b-a} \int_{a}^{b} v(t) d t $$ Are the "average velocity on the interval \([a, b]^{\prime \prime}\) and the "average value of the velocity function on \([a, b]^{\prime \prime}\) the same thing? Why or why not? Explain.

Let \(f(x)=3-x^{2}\) and \(g(x)=2 x^{2}\) a. On the interval [-1,1] , sketch a labeled graph of \(y=f(x)\) and write a definite integral whose value is the exact area bounded by \(y=f(x)\) on [-1,1] . b. On the interval \([-1,1],\) sketch a labeled graph of \(y=g(x)\) and write a definite integral whose value is the exact area bounded by \(y=g(x)\) on [-1,1] . c. Write an expression involving a difference of definite integrals whose value is the exact area that lies between \(y=f(x)\) and \(y=g(x)\) on [-1,1] . d. Explain why your expression in (c) has the same value as the single integral \(\int_{-1}^{1}[f(x)-\) \(g(x)] d x\) e. Explain why, in general, if \(p(x) \geq q(x)\) for all \(x\) in \([a, b],\) the exact area between \(y=p(x)\) and \(y=q(x)\) is given by $$ \int_{a}^{b}[p(x)-q(x)] d x $$

Filters at a water treatment plant become dirtier over time and thus become less effective; they are replaced every 30 days. During one 30 -day period, the rate at which pollution passes through the filters into a nearby lake (in units of particulate matter per day) is measured every 6 days and is given in the following table. The time \(t\) is measured in days since the filters were replaced. $$ \begin{array}{lcccccc} \hline \text { Day, } t & 0 & 6 & 12 & 18 & 24 & 30 \\ \hline \text { Rate of pollution in units per day, } p(t) & 7 & 8 & 10 & 13 & 18 & 35 \\ \hline \end{array} $$ Table 4.1.12: Pollution data for the water filters. a. Plot the given data on a set of axes with time on the horizontal axis and the rate of pollution on the vertical axis. b. Explain why the amount of pollution that entered the lake during this 30 -day period would be given exactly by the area bounded by \(y=p(t)\) and the \(t\) -axis on the time interval [0,30] . c. Estimate the total amount of pollution entering the lake during this 30 -day period. Carefully explain how you determined your estimate.