Project #4456 - Math Project

Willing to pay up to $50 if work is done quickly. Thank you. 

 

 

Part 1

a. On graph paper, sketch the graph of y=sin x. Draw the scecant line joining (0,0) and (2pi/3,2pi/3). Find the slope of this line. Is the slope of the tangent line at (0,0) greater than or less than the slope of this secant line?

 

b. In the same graph as a) draw the secant line joining (0,0) and (pi/2,sinpi/2). Find the slope of this line. Is the slope of the tangent line at (0,0) greater than or less than the slope of this secant line?

 

c. In the same graph as a) draw the secant line joining (0,0) and (pi/4,sinpi/4).  Is the slope of the tangent line at (0,0) greater than or less than the slope of this secant line?

 

d. Find the slope of the line joining (x, f(x)) and (x+h, f(x+h)) in terms of the nonzero number h. Taking x=0, verify that h= 2pi/3, pi/2, pi/4 yield the slopes you caluculated in part b-d.

 

 

Part 2

1a. Sketch the graph of f(x) = 2^x

b. Shift the graph of f(x) to the left by 3 unites. Write a new function, g(x), in terms of f(x), and sketch the resulting function on the same grid as a). 

c. Compress your result from b) vertically by a factor of 1/4. Write the new function, g(x), in terms of f(x), and sketch the resulting transformation on the same grid as a).

d. Reflect the result from c) across the x-axis. Write the new function, g(x), in terms of f(x), and sketch the resulting transformation on the same grid as a). 

e. You now have an equation for g(x) in terms of f(x). Write an equation for the function g(x) in terms of x.

2.

a. Sketch the graph of f(x) = 2^x

b. Shift the graph of f(x) to the left by 1 unit. Write a new function, h(x), in terms of f(x), and sketch the resulting transformation on the same grid as a).

c. Reflect the result from b) across the x-axis. Write the new funciton, h(x), in terms of f(x), and sketch the resulting transformation on the same grid as a).

d. You now have an equation for h(x) in terms of f(x). Write an equation for the function h(x) in terms of x.

3. Take the equation for g(x) from 13) and the equation for h(x) from 2d) and show algebraically that g(x) = h(x)

 

Part 3

1.Given f(x) = 2^x, find the average rate of change of f(x) from x=0 to x= 0.1, so that x= .1. Recall that the average rate of change is y/x. Show steps.

2. Continue finding the avg. rate of change using 0 as your inital x-value and letting x get smaller. Write the results in a table.

Interval                           Avg Rate of change

[0,0.1]

[0,0.01]

[0,0.001]

[0,0.0001]

 

3. Using your observations from the table, complete this sentence on your paper: As x gets smaller and smaller, y/x gets ____________________.

4. Make a similar table for g(x) = 3^x, using the same intervals, and write down.

5. Using your onservations from this new table, complete this sentence on your paper: As x gets smaller and smaller, y/x gets _________________.

6. For both functions it seems that as x gets smaller, the avg rate of fhange approaches a particular decimal. One of the decimals is greater than 1 and the other is less than 1. Using trial and error, try to find an exponential function whose average rate of change, using the same intervals of above, approaches 1. You should present at least 3 other exponential functions and their tables and each table should be accompanied by an observation similar to those in parts 3 and 5.

7. Write your conjecture about the exponential funciton whose average rate of change on the interval [0,x], as x gets smaller and smaller, comes the closest to 1.

 

Subject Mathematics
Due By (Pacific Time) 04/16/2013 04:00 pm
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