Assignment 4a

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course mth 163

6/4 1pm

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Question: `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.

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Your solution:

f(3) = 3^2+4 = 13 : f(7) = 7^2+4 = 53 : f(-5) = (-5)^2+4 = 29

The points on the graph will be (3, 13), (7, 53), and (-5, 29) The graph is a parabola facing down.

confidence rating #$&*: 3

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Given Solution:

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.

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Self-critique (if necessary): I do not understand how the question is asking for all the information that is in the solution.

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Self-critique rating: 3

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The question asked for a description of the graph.

Your description was good.

It is expected that you will also do your best to understand everything in the given solution.

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Question: `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.

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Your solution:

F(a) = a^2+a

F(x+2) = (x+2)^2+4 = x(x+2)+2(x+2)+4 = x^2+2x+2x+4+4 = x^2+4x+8

F(x+h) = (x+h)^2+4 = x^2+2hx+h^2+4

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Not bad, but this would be

F(x+h) - f(x) = x^2+2hx+h^2+4 - (x^2+4)

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= 2hx+h^2

confidence rating #$&*: 2

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Given Solution:

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.

You should have written these expressions out, and the following should probably be represented on your paper in form similar to that given here:

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Self-critique (if necessary): I was confused on how to write the last one.

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Self-critique rating: 3

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Question: `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.

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Your solution:

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) / (x2-x1)

confidence rating #$&*: 2

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Given Solution:

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.

Compare what you have written down with the expressions below:

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Self-critique (if necessary): I not clear on what a subscript is? I worked the problem but will I have to know how to use them.

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Self-critique rating: 3

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x1, for example, will be understood to stand for x with subscript 1. That is, x written with a small 1 to its right and slightly below.

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Question: `q004. If f(x) = 5x + 7, then for what value of x is f(x) equal to -3?

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Your solution:

5x+7 = -3, 5(-2)+7 = -3

confidence rating #$&*: 3

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Given Solution:

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.

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Self-critique (if necessary):

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&#Good responses. See my notes and let me know if you have questions. &#