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Run Multiple Choice Generator at h:\shares\physics
Run Core Problems by giving the program qzz as your 'course'--enter this when asked for your 3-digit course number.
Do Precalculus Quiz 1018

Work the Major Quiz problems below prior to Monday's class.

Work through the worksheet Introduction to Exponential Functions listed for Assignment 15, down through Exercise 7.

For the function f(x) = x^2 - 2 x + 3, evaluate at x = 2, 4, 6, 8 and write your results as a sequence.

Substituting we get f(2), f(4), f(6) and f(8) equal to 3, 11, 27, 51.

Analyze the sequence to find its pattern.

The sequence

3, 11, 27, 51, ...

has first differences

8, 16, 24

and second differences

8, 8.

If we extend the second-difference sequence we get

8, 8, 8, 8, ...

The first-difference sequence will then extend to give us new members 24+8 = 32 and 32 + 8 = 40, etc.. So the first-difference sequence is

8, 16, 24, 32, 40, ...

The original sequence extends by adding the members of the first-difference sequence to get 51 * 32 = 83 and 83 + 40 = 123, etc.. The original sequence therefore extends as

3, 11, 27, 51, 83, 123, ... .

For the function f(x) = x^2 - 2 x + 3, what is the average slope between the x = x1 and x = x2 points?

The graph points are (x1, f(x1)) and (x2, f(x2)).

The slope between these two points is

slope = rise / run = change in y / change in x = ( f(x2) - f(x1) ) / (x2 - x1).

The values of the function are

f(x1) = x1^2 - 2 x1 + 3 and
f(x2) = x2^2 - 2 x2 + 3

so our slope expression ( f(x2) - f(x1) ) / (x2 - x1) becomes

slope =

[ x2^2 - 2 x2 + 3 - ( x1^2 - 2 x1 + 3) ] ( x2 - x1) =

[ x2^2 - 2 x2 + 3 - x1^2 + 2 x1 - 3 ] / (x2 - x1) =

[ ( x2^2 - x1^2 ) - 2 ( x2 - x1) ] / ( x2 - x1) =

[ ( x2^2 - x1^2 ) / ( x2 - x1) ] - [ 2 ( x2 - x1) / ( x2 - x1) ] =

[ ( x2 - x1)(x2 + x1) / ( x2 - x1) ] - [ 2 ( x2 - x1) / ( x2 - x1) ] =

(x2 + x1) - 2

Problem Number 1

Explain what the number e represents.

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.

At what average rate does the depth change during each time interval?

Sketch a graph of depth vs. clock time.

Use a sketch to explain why the slope of this graph between 54.8 and 82.2 sec represents the average rate at which depth changes during this time interval.

Problem Number 3

We expect that one of the power functions y = a x^.5, y = a x^-.5, y = a x^2 and y = a x^-2 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?

Problem Number 4

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

Substitute the coordinates of the points into the form of a straight line and solve the equations for the appropriate parameters.
Use the slope = slope form, complete with explanation and a labeled graph.

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

What does the slope of the graph represent?
What is the length when the weight is 9 pounds?
What is the weight when the spring length is 13 cm?

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.

Problem Number 6

Solve using ratios instead of functional proportionalities:

If a sand pile 2.3 meters high has a mass of 4866.8 kg, then what would we expect to be the mass of a geometrically similar sand pile 4.7 meters high?
If there are 1.7457 billion grains of sand exposed on the surface of the first sand pile, how many grains of sand we expect to be exposed on the surface of the second?

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)?