1022
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
The values of the function are
so our slope expression ( f(x2) - f(x1) ) / (x2 - x1) becomes
[ 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.
Sketch a graph of depth vs. clock time.
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:
If your graph represents the length of a spring, in cm, vs. hanging weight in pounds:
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:
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)?