Assignment 4 Questions

course Mth 163

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(x) = x^2 + 4

f(3)= 3^2 +4

f(3)= 9+4= 13

f(x) = x^2 + 4

f(7)= 7^2 +4

f(7)= 49+4= 53

f(x) = x^2 + 4

f(-5) = -5^2 +4

f(-5) = 25+4 = 29

The coordinates are as follows:

(3, 13)

(7, 53)

(-5, 29)

The graph is a parabola with the above coordinates included. The vertex happens to be (0, 4).

Confidence Assessment: 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):

Self-critique Rating:

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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 + 4

f(x+2) = x+2^2 + 4

f(x +2) = x^2 + 4x + 4 +4

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

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

f(x+h) - f(x) ] / h = (x+h) - f(x) ] / h ^2 + 4

f(x+h) - f(x) ] / h = x^2] / h ^2 + 4

Couldn’t figure this one out.

Confidence Assessment: 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

( 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.

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Self-critique (if necessary):

I couldn’t figure the hardest one (last one) out. It was very confusing.

You need to write the expressions out carefully on paper.

Self-critique Rating: 2

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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(x) = 5x + 7

f= 5(x^1) +7

f(x) = 5x + 7

f= 5(x^2) +7

Confidence Assessment: 1

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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.

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Self-critique (if necessary):

I can get the easy ones but I don’t understand how the solution for the harder ones is done. I don’t know what you multiply to get the answer.

I'll be glad to clarify but I need to know what you are seeing. Can you be very specific about what step or steps you do not understand in the given solution?

( 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)

Self-critique Rating:

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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:

f(x) = 5x + 7

f(x) = 5x + 7

-3= 5x +7

-10= 5x

x= -2

Confidence Assessment: 3

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&#Your work looks good. See my notes. Let me know if you have any questions. &#