.................................................
......!!!!!!!!...................................
00:10:57 `q001. Note that this assignment has 4 questions
If f(x) = x^2 + 4, then find the values of the following: f(3), f(7) and f(-5). Plot the corresponding points on a graph of y = f(x) vs. x. Give a good description of your graph.......!!!!!!!!...................................
RESPONSE --> If we have the equation f(x) = x^2 +4 then
f(3) = 3^2 + 4 f(3) = 9 + 4 f(3) = 13 f(7) = 7^2 + 4 f(7) = 49 + 4 f(7) = 53 f(-5) = -5^2 + 4 f(-5) = 25 + 4 f(-5) = 29 I am unsure of the model I have developed..................................................
......!!!!!!!!...................................
00:14:39 f(x) = x^2 + 4. To find f(3) we replace x by 3 to obtain
f(3) = 3^2 + 4 = 9 + 4 = 13. Similarly we have f(7) = 7^2 + 4 = 49 + 4 = 53 and f(-5) = (-5)^2 + 9 = 25 + 4 = 29. Graphing f(x) vs. x we will plot the points (3, 13), (7, 53), (-5, 29). The graph of f(x) vs. x will be a parabola passing through these points, since f(x) is seen to be a quadratic function, with a = 1, b = 0 and c = 4. The x coordinate of the vertex is seen to be -b/(2 a) = -0/(2*1) = 0. The y coordinate of the vertex will therefore be f(0) = 0 ^ 2 + 4 = 0 + 4 = 4. Moving along the graph one unit to the right or left of the vertex (0,4) we arrive at the points (1,5) and (-1,5) on the way to the three points we just graphed.......!!!!!!!!...................................
RESPONSE --> I have made notes on this in my notebook. I was unsure as to if I had graphed the correct coordinates.
.................................................
......!!!!!!!!...................................
00:24:07 `q002. If f(x) = x^2 + 4, then give the symbolic expression for each of the following: f(a), f(x+2), f(x+h), f(x+h)-f(x) and [ f(x+h) - f(x) ] / h. Expand and/or simplify these expressions as appropriate.
......!!!!!!!!...................................
RESPONSE --> If f(x) = x^2 + 4 then we get the following through substitution
f(a) = a^2 +4 f(x+2) = (x + 2)^2 + 4 f(x+2) = (x+2)(x+2) + 4 f(x+h) = (x + h)^2 + 4 f(x+h) = (x+h)(x+h) +4 f(x+h) - f(x) = (x+h)^2 + 4 - x^2 + 4 f(x+h) - f(x) = (x+h)(x+h) - x^2 + 8 f(x+h) - f(x) = x^2 + xh + xh + h^2 - x^2 +8 f(x+h) - f(x) = 2xh + h^2 + 8 [f(x+h) - f(x)] / h We found above that f(x+h) - f(x) is equal to 2xh + h^2 +8 So 2xh + h^2 + 8 / h We can eliminate the ""h"" so that 2x + h^2 + 8.................................................
......!!!!!!!!...................................
00:25:15 If f(x) = x^2 + 4, then the expression f(a) is obtained by replacing x with a:
f(a) = a^2 + 4. Similarly to find f(x+2) we replace x with x + 2: f(x+2) = (x + 2)^2 + 4, which we might expand to get (x^2 + 4 x + 4) + 4 or x^2 + 4 x + 8. To find f(x+h) we replace x with x + h to obtain f(x+h) = (x + h)^2 + 4 = x^2 + 2 h x + h^2 + 4. To find f(x+h) - f(x) we use the expressions we found for f(x) and f(x+h): f(x+h) - f(x) = [ x^2 + 2 h x + h^2 + 4 ] - [ x^2 + 4 ] = x^2 + 2 h x + 4 + h^2 - x^2 - 4 = 2 h x + h^2. To find [ f(x+h) - f(x) ] / h we can use the expressions we just obtained to see that [ f(x+h) - f(x) ] / h = [ x^2 + 2 h x + h^2 + 4 - ( x^2 + 4) ] / h = (2 h x + h^2) / h = 2 x + h.......!!!!!!!!...................................
RESPONSE --> ok
.................................................
......!!!!!!!!...................................
00:33:30 `q003. If f(x) = 5x + 7, then give the symbolic expression for each of the following: f(x1), f(x2), [ f(x2) - f(x1) ] / ( x2 - x1 ). Note that x1 and x2 stand for subscripted variables (x with subscript 1 and x with subscript 2), not for x * 1 and x * 2. x1 and x2 are simply names for two different values of x. If you aren't clear on what this means please ask the instructor.
......!!!!!!!!...................................
RESPONSE --> Given that f(x) = 5x + 7 we can substitute to get symbolic expressions for the following:
f(x1) = 5(x1) + 7 f(x2) = 5(x2) + 7 [f(x2) - f(x1)] / (x2 - x1) [5(x2) + 7 - 5(x1) + 7] / (x2 - x1) 5(x2) - 5(x1) +14 / (x2 - x1).................................................
......!!!!!!!!...................................
00:35:57 Replacing x by the specified quantities we obtain the following:
f(x1) = 5 * x1 + 7, f(x2) = 5 * x2 + 7, [ f(x2) - f(x1) ] / ( x2 - x1) = [ 5 * x2 + 7 - ( 5 * x1 + 7) ] / ( x2 - x1) = [ 5 x2 + 7 - 5 x1 - 7 ] / (x2 - x1) = (5 x2 - 5 x1) / ( x2 - x1). We can factor 5 out of the numerator to obtain 5 ( x2 - x1 ) / ( x2 - x1 ) = 5.......!!!!!!!!...................................
RESPONSE --> I failed to factor out the 5 from the final equation.I am not sure that I understand why the 7 in the f(x1) equation became a negative in the second step of the equation.
.................................................
......!!!!!!!!...................................
00:40:26 `q004. If f(x) = 5x + 7, then for what value of x is f(x) equal to -3?
......!!!!!!!!...................................
RESPONSE --> First we must set up the equation to read
3 = 5x + 7 We can solve this equation by subtracting 7 from both sides to get -10 = 5x which we can solve by dividing both sides by 5 to get x = -2 We would then substitute the -2 into the equation to test it. f(-2) = 5(-2) + 7 f(-2) = -10 + 7 f(-2) = 3 so we can see that with the value of -2 as the f(x) will give us -3 as an answer..................................................
......!!!!!!!!...................................
00:40:33 If f(x) is equal to -3 then we right f(x) = -3, which we translate into the equation
5x + 7 = -3. We easily solve this equation (subtract 7 from both sides then divide both sides by 5) to obtain x = -2.......!!!!!!!!...................................
RESPONSE --> ok
.................................................
ޚw黭avϲx Student Name: assignment #003
.................................................
......!!!!!!!!...................................
21:53:29 `q001. Note that this assignment has 6 questions
The function y = a x^2 + b x + c takes the value y = 0 when x = [ -b + `sqrt(b^2 - 4 a c ] / (2 a) or when x = [ -b - `sqrt(b^2 - 4 a c ] / (2 a). For the function y = - 0.45833 x^2 + 5.33333 x - 6.875, which you obtained as a quadratic model of the points (1, -2), (3, 5) and (7, 8) in the preceding assignment, find the values of x for which y = 0. Compare to the estimates you made from the graph through (1,-2), (3, 5) and (7, 8) in Assignment 1.......!!!!!!!!...................................
RESPONSE --> First we would substitute the values for a, b and c into the equation to get
y = [-5.33333 + 'sqrt(5.33333^2) - 4(0.45833)(-6.875) / 2(0.45833) y = [-5.33333 + 'sqrt(5.33333) + 12.604075 / 0.91666 y = [-5.33333 + 2.3094003 + 12.604075 / 0.91666 y = 9.580146 / 0.91666 y = 10.451144 or 10.45 +/- 10.45.................................................
......!!!!!!!!...................................
21:54:12 09-18-2005 21:54:12 For the function y = - 0.45833 x^2 + 5.33333 x - 6.875 we have a = -0.45833, b = 5.33333 and c = -6.875. The quadratic formula therefore tells us that for our function we have y = 0 when
x = [-5.33333 + `sqrt(5.33333^2 - 4 * (-0.45833 ) * (-6.875)) ] / ( 2 * (-0.45833)) = 1.47638 and when x = [-5.33333 - `sqrt(5.33333^2 - 4 * (-0.45833 ) * (-6.875)) ] / ( 2 * (-0.45833)) = 10.16006.......!!!!!!!!...................................
NOTES -------> I apparently did not understand the quesiton, I am making notes here though.
.......................................................!!!!!!!!...................................
21:55:30 `q002. Extend the smooth curve in your sketch to include both points at which y = 0. Estimate the x value at which y takes its maximum value.
......!!!!!!!!...................................
RESPONSE --> ?
.................................................
......!!!!!!!!...................................
21:56:27 Your graph should clearly show how the parabola passes through the x axis at the points where x is approximately 1.5 (corresponding to the more accurate value 1.47638 found in the preceding problem) and where takes is a little more than 10 (corresponding to the more accurate value 10.16006 found in the preceding problem).
The graph of the parabola will peak halfway between these x values, at approximately x = 6 (actually closer to x = 5.8), where the y value will be between 8 and 9.......!!!!!!!!...................................
RESPONSE --> Making these notes as well since I had incorrect information from the first problem, it was not possible to get this one correct either.
.................................................
"