Paul runs to catch the bus along a line in such a way that at time t, 1 ≤ t ≤ 8, his position is given by:

(a) Write a formula for the velocity of Paul at time t.

(b) At what instant does Paul reach his maximum speed.

(c) When is Paul moving to the left?

(d) Find the total distance traveled by Paul from t = 1 to t = 8.

Quick Find Code For This Problem.

Legend: [QUES6LGND]

Solution For Part a.): [QUES6PA]

Solution For Part b.): [QUES6PB]

Solution For Part c.): [QUES6PC]

Solution For Part d.): [QUES6PD]

How To Use Quick Find.

-To Use Quick Find, copy (Ctrl+C/Command+C) the code (All the stuff INSIDE the square brackets in the code region above) for which ever section you'd like to go to, then, hit in this order;

Ctrl+F

Ctrl+V

Enter/Return

or if your on a Mac You'd hit

Command+F

Command+V

Enter/Return.

Then, your webbrowser should bring you directly to the beginning of the solution indicated by the code, making navigation through longish solutions much easier. Enjoy :]

**[QUES6LGND]**

__Legend__

Blue: Side Notes or "Paul says"

Green: Terms and Definitions

Red: Final Answers

The four permissible (TI-83) calculator functions that we're allowed to use to solve these wiki questions:

- Graphing the function: [Y=] [GRAPH]
- Determining the roots or y-intercepts: [2nd] [CALC] [2:zero]
- Numerically differentiating a function: [2nd] [6:dy/dx] or [MATH] [8:nDeriv]
- Numerically integrating a function: [2nd] [7:∫f(x)dx] or [MATH] [9:fnInt]

**[QUES6PA]**

**PART A**

Differentiating a position function results in a velocity function. Symbolically, v(t) = s’(t).

In this case, we're differentiating an antiderivative. (Paul says: Note the "∫".) Differentiation and antidifferentiation can be seen as inverse processes of each other: The derivative of x^{2} is 2x; an antiderivative of 2x is x^{2}.

Not only are we differentiating an antiderivative, we’re differentiating an *accumulation function*, a function that measures the accumulating area under a graph.

The derivative of an accumulation function is the original function, by *The Second Fundamental Theorem of Calculus*.

We can see there is a function within a function in s(t). (Paul says: Note there are two variables, t and x.)

To differentiate a function within a function, we use *The Chain Rule*.

The velocity function or the derivative of the position function is:

**[QUES6PB]**

**PART B**

To determine Paul’s maximum speed, we first need to determine the extrema of v(t).

*Velocity* is speed with direction. *Speed* doesn't have direction. So, the absolute value of velocity is speed--all values of v(t) must be positive. To have a negative velocity means Paul is going in a direction opposite to what we designated positive. By convention, right is designated positive and left is designated negative.

We’re limited to using those four permissible calculator functions, so we can’t use the calculator’s maximum function. We’ll have to use a different approach: finding the roots of v'(t)!

To differentiate products, we use *The Product Rule*:

Paul says: Wherever v’(t) = 0 (the roots/the zeroes/the y-intercepts), we know v(t) has a *local extrema* (minimum or maximum)!

Since v(t) is a *transcendental function*, a function that contains an exponential function and a trigonometric function, we can’t apply the algebra we know to solve for the roots of v’(t), so we have to use our calculator.

Using the calculator, v’(t) = 0 at t = 1.2340473, 3.7311818, and 6.7002108. This can be seen on the graph of v'(t).

The graph of v'(t).

Note: Because the graphing program used has its limitations, the domain from 0 < t < 9 is shown. The domain of v(t), as indicated in the problem, is 1 ≤ t ≤ 8.

But, don’t forget all this happens in a closed interval, and, by *The Extreme Value Theorem*, the endpoints of a function are also considered as local extrema.

Paul says: The endpoints are at t = 1 and t = 8, since the domain is 1 ≤ t ≤ 8!

- v(1.2340473) = 0.39375796
- v(3.7311818) = 4.833368255
- v(6.7002108) = -5.89643534
- v(1) = 0.45969769
- v(8) = 0.10668763

Therefore, at t = 6.7002108, Paul is at his maximum speed, which is 5.89643534.

Paul says: t = 6.7002108 is a minimum on the graph of v(t). But, in the context of the problem, the question asked is to determine the maximum speed. A negative velocity indicates running in the direction opposite to what we designated as positive. So, 5.90 m/s [left] = -5.90 m/s!

**[QUES6PC]**

**PART C**

When the v(t) is negative, Paul is moving to the left. So, we need to know where the roots of v(t) are because the roots indicate a change in sign, positive to negative or negative to positive.

Using the calculator function to determine roots on the graph, we find v(t) = 0 at t = 5.2035697 and t = 7.987021 in 1 ≤ t ≤ 8.

The graph of v(t).

Note: Because the graphing program used has its limitations, the domain from 0 < t < 9 is shown. The domain of v(t), as indicated in the problem, is 1 ≤ t ≤ 8.

Looking at the graph, we can deduce that v(t) is negative at 5.2035697 < t < 7.987021.

Or, instead of using a graph, we could use a *line analysis*.

- By plugging a number to the left of 5.2035697, we obtain a positive number.
- By plugging a number to the right of 5.2035697 or to the left of 7.987021, we obtain a negative number.
- By plugging a number to the right of 7.987021, we obtain a positive number.

We see that v(t) is negative (Paul is moving to the left) at 5.2035697 < t < 7.987021.

**[QUES6PD]**

**PART D**

Integrate, from t = 1 to t = 8, the absolute value of the velocity function to get distance.

The graph of v(t) integrated from t = 1 to t = 8.

Note: Because the graphing program used has its limitations, the domain from 0 < t < 9 is shown. The domain of v(t), as indicated in the problem, is 1 ≤ t ≤ 8.

Paul says: If we don’t find the absolute value, the positive areas and the negative areas will cancel each other, and our final answer would be the displacement!

*Distance* can be defined as how much ground Paul walked, ignoring his direction.

*Displacement* can be defined as the change in position, where his direction is indicated.

Using the *function numerical integrator (fnInt)* in our calculator, the integral of the absolute value of the velocity function at 1 ≤ t ≤ 8 is 21.46124296.

Paul says: NEVER PUT THIS ON THE AP EXAM!

Paul says: PUT THIS ON THE EXAM INSTEAD!

Therefore, Paul ran 21.46124296 units in total.