Assignment 4

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course Mth 163

Question: `q001. Note that this assignment has 7 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:

If f(x) = 3 then we get 3^2 + 4; multiply

9 + 4; add

13

If f(x) = 7 we get 7^2 + 4; multiply

49 + 4 ; add

53

If f(x) = -5 we get -5^2 + 4; multiply

25 + 4 ; add

29

When we go to make a graph we plot these points (3,13) (7,53) (-5,29). To find the vertex we have to take the quadratic formula for f(x) where a = 1 b = 0 c = 4 and plug it into the equation -b / (2a)

So therefore we get

-0 / 2*1 = 0 so x = 0 to find y we plug in f(0) into the equation.

F(0)= 0^2 +4

0+4

4 so y = 4

The vertex of the parabola is (0,4).

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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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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:

Replace x with a:

F(a) = a^2 + 4

Replace x with x+2

x^2 + 4x +8

Replace x with x + h

x^2 + 2hx + h^2 + 4

Replace x with (x+h) -x

(x^2 + 2hx + h^2 + 4) - (x^2 +4)

2hx + h^2

Replace x with (x+h)-(x) / h

(x^2 + 2hx + h^2 +4) - (x^2 + 4) / h

2hx + h^2 / h

2hx + h

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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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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 = 5x1 + 7

f(x2) = 5* x2 + 7 = 5x2 + 7

f(x2)- f(x1) / (x2-x1) =

(5* x2 + 7) - (5*x1 + 7) / (5*x2+7 - 5*x1+7)

(5x2 - 5x1) / (x2 - x1)

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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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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:

If f(x) = -3 then we simply plug that into our equation

5x + 7 = -3 ; subtract 7

5x = -10 ; divide by 5

x = -2

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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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Question: `q005. If f(x) = 3 x + 2 then what are the values of f(3),f(x+3), 3 f(x) and f(x+h) - f(x)?



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Your solution:

f(3) = 3 * 3 + 2 = 11

f(x+3) = 3(x+3) + 2 = 3x + 9 + 2 = 3x+11

f(x+h) - f(x) = 3(x+h) - (3x+2) = 3x + 3h - 3x + 2 = 3h +2

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If f(x) = 3 x + 2, then f(x + h) = 3 ( x + h ) + 2, not just 3 ( x + h) .

-(3 x + 2) = -3x - 2, not -3x + 2.

The result here will be just 3 h. Be sure you see why. Use the Question Form if you're not sure.

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Question: `q006. If f(x) = 3 x + 2 then what is the value of f(0)? Faor what value(s) of x do we have f(x) = 0?

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Your solution:

f(0) = 3 * 0 + 2 = 2

f(x) =0

3x + 2 = 0 : subtract by 2

3x = -2 ; divide by 3

x= -2/3

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Question: `q007. Evaluate the expression

• (f(b) - f(a)) / (b - a)

for the function f(x) = 2 ( x - 3 ) + 5.

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Your solution:

f(x) = 2 ( x-3) + 5 = 2x -1

f(b)= 2 * b -1 = 2b-1

f(a)= 2 * a -1 = 2a -1

f(x) = (2 * 2b -1) - (2 * 2a -1) / (2b-1-2a-1)

= (4b-1)-(4a-1)/ 2b-2a

= 4b - 4a / 2b - 2a

= 2b-2a

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Good to here.

It's not clear how you were thinking in the next step, but at this point you would simply replace f(b) by 2 b - 1 and f(a) by 2 a - 1 to get

( (2 b - 1) - (2 a - 1) ) / (b - a).

If you simplify this you get just 2.

Use a question form if you don't see this and can't get the expression to simplify.

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Good overall, but you do have some errors. Check my notes and respond as appropriate.

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