1025

Compounding $1000 initial principle continuously, for 1 year, at 100% annual interest:

Practice Test 1

Problem Number 1

Explain what the number e represents.

e is the factor by which money grows in a year if we compound annual interest at 100%, compounding continuously, for 1 year.

This factor is the limiting value of (1 + 1 / n)^ n as n approaches infinity.

If we compound for t years the factor is e^t, so that if our initial principle is P0 our principle after t years is P(t) = P0 * e^t.

If we were to compound for t years at interest rate r the principle function would be P(t) = P0 * e^(rt).

Problem Number 2

If f(x) = x2, what are the vertex and the three basic points of the graphs of f(x- .75), f(x) - .35, 5 f(x) and 5 f(x- .75) + .35. Quickly sketch each graph.

At clock times 54.8, 82.2, 109.6 and 137 sec, we observe water depths of -59, -95.4, -116.8 and -123.3 cm.

Sketch a graph of depth vs. clock time.

This is Major Quiz material.  We won't address it today, but be sure you review.

Problem Number 3

We expect that one of the power functions

best fits the following data: 

When x takes values

15.448, 21.737, 26.758 and 31.304 ,

 respectively, y takes values

5, 10, 15 and 20. 

Which power function best fits this data?

As shown in the worksheets, substitute x = 15.45 and y = 5 into the y = a x^.5 and get a.  Then substitute, say, the last point x = 31.3 and y = 20.  If you get very nearly the same value of a then try the other two points.  If they all give you about the same value of a then you have your model.

Otherwise repeat for y = a x^-.5.

You can also use some common sense.  It looks like y values increase with increasing x values.  This eliminates two of the possibilities.

If you know how the basic power functions behave (e.g., basic points), increasing or decreasing at an increasing or decreasing rate, then you can pretty much spot which function to try first.

Problem Number 4

Find the equation of a line through ( 3, 9) and ( 5, 8) by each of the following methods:

If your graph represents the length of a spring, in cm,  vs. hanging weight in pounds:

The form of a straight line is y = a x + b.  Substituting the points we get the equations

We can solve to get our values of a and b.  We get a = -.5  and b = 10.5  so our model is   

We can then use the slope = slope form.  Placing the two points and the general point (x,y) on a graph we find that the slope of the line is (8 - 9) / (5 - 3).  The slope from the first point to (x, y) is (y - 9) / (x - 3).  Setting the two slopes equal we get

which we can solve for y to get

Problem Number 5

Explain how to use two simultaneous linear equations, obtained from two given points, to obtain the equation of the line through the two points.

See above.

Problem Number 6

Solve using ratios instead of functional proportionalities: 

The proportionality equation for volume is y = k x^3.  Since mass occupies volume we use this proportionality.

This proportionality tells us that y2 / y1 = k x2^3 / ( k x1^3) = (x2 / x1)^3.

So mass2 / mass1 = (height2 / height1) ^ 3 = (4.7 m / 2.3 m)^3 = 8.5 so

mass2 = 8.5 * mass1 = 8.5 * 4867 kg = etc..

Sand grains are exposed at the surface.  The proportionality for surface areas is y = k x^2.  The same sort of analysis as used above tells us that the ratios are

area2 / area1 = (height2 / height1)^2 so we get

...

Problem Number 7

Show that the slopes of the function y = .9 t^2 + -30 t + -75 change at a constant rate.

.Problem Number 8

Problem:  Obtain a quadratic depth vs. clock time model if depths of 67.75163 cm, 49.07069 cm and 38.95717 cm are observed at clock times t = 15.01528, 30.03055 and 45.04583 seconds.

Problem: The quadratic depth vs. clock time model corresponding to depths of 67.75163 cm, 49.07069 cm and 38.95717 cm at clock times t = 15.01528, 30.03055 and 45.04583 seconds is depth(t) = .019 t2 + -2.1 t + 95. Use the model to determine the clock time at which depth is 54.61883 cm.

 

Problem Number 4

If y = -.4 t^2 + -6 t + 66, what symbolic expression stands for the slope between the graph points for which t = x and t = x+h?

Problem Number 7

If a(n) = a(n-1) + 9, with a(0) = -9, then what is the value of a( 310)?