open qa 28

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course Mth 271

Question: `qQuery problem 1 d 7th edition 3.7.12 sketch y = -x^3+3x^2+9x-2

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Describe your graph, including a description of any intercepts, extrema, asymptotes and concavity.

Your solution:

Zeros to test are +-1, +-2

Test shows that the zero is -2

Factor is (x+2)

By long division: -x^2 + 5x -1 or x^2 -5x +1

(x+2)(x^2 - 5x +1)

X = -2

And using quadratic we have two solutions:

-5 +sqrt(21)/2 = -0.21

-5 - sqrt(21)/2 = -4.79

Y intercept is y = 2

F’(x) = -3x^2 + 6x + 9 = 3 (x+1) (x-3)

X = -1, x = 3 are critical points

Intervals: (- infinity, -1), (-1, 3), and (3, infinity)

First interval < 0 Function is decreasing

Second interval > 0 Function is increasing

Function has a minimum at -1.

Third interval function is increasing.

F”(x) = -6x + 6 = 6 (-x + 1)

X = 1

X = 1 is point of inflection

(- infinity, 1), (1, infinity)

First interval concave upwards. Second interval concave downwards.

confidence rating #$&*:

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Given Solution:

`a First we find the zeros:

You can find the zero at x = -2 by inspection (i.e., try a few simple values of x and see if you 'hit' one).

Knowing that x = -2 is a zero, you know that (x - (-2) ) = x + 2 is a factor of the expression. You can find the other factor by long division of x + 2 into -x^3 + 3 x^2 + 9x - 2. You get quotient x^2 - 5 x + 1.

Thus -x^3 + 2 x^2 + 9 x - 2 = - (x + 2)•(x^2 - 5•x + 1).

This expression is zero when x+2 = 0 or when x^2 - 5 x + 1 = 0.

The first equation we already know gives us x = -2.

The second is solved by the quadratic formula. We get

x = [ -(-5) +- `sqrt( (-5)^2 - 4 * 1 * 1 ) ] / (2 * 1) = [ 5 +- `sqrt(21) ] / 2.

Simplifying we get approximate x values .21 and 4.7.

Then we find maxima and minima using 1st and 2d derivative:

The derivative is -3x^2+6x + 9, which gives the equation -3x^2+6x + 9 = 0 for critical points. Dividing thru by -3 we get

x^2 - 2x - 3 = 0 or(x-3)(x+1) = 0 so

x = 3 or x = -1.

Second derivative is -6x + 6, which is negative when x = 3 and positive when x = -1.

At x = 3 we have a maximum. Evaluating y = -x^3+3x^2+9x-2 at x = 3 we get y = 25. The negative second derivative indicates that (3,25) is a maximum.

At x = -1 we have a minimum. Evaluating y = -x^3+3x^2+9x-2 at x = -1 we get y = -7. The positive second derivative indicates that this (-1,-7) is a minimum.

Finally we analyze 2d derivative for concavity and pts of inflection:

The second derivative -6x + 6 is zero when x = 1; at this point the derivative, which is linear in x, changes from positive to negative. Thus the x = 1 point (1, 9) is a point of inflection.

The derivative is positive and the function therefore concave up on (-infinity, 1).

The derivative is negative and the function therefore concave down on (1, infinity).

The function is defined for all x so there are no vertical asymptotes.

As | x | -> infinity the magnitude of the function -> infinity so there are no horizontal asymptotes. **

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Self-critique (if necessary):

The only thing I misses is the evaluation of maximum with 2nd derivative. Will try to remember that one, although it is a little bit confusing.

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Self-critique Rating:

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Question: `qQuery problem 1 g 8th edition 3.7.34 7th edition 3.7.32 sketch (x^2+1)/(x^2-1)

Note: The problem in the text might be (x^2+1)/(x^2-2). If so the solution given below can be easily adapted to that function.

Describe your graph, including a description of any intercepts, extrema, asymptotes and concavity.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

X intercepts:

X^2 +1 = 0

X^2 = -1 there are no x intercepts

Y intercepts: y = -1

Horizontal asymptote is y = 1

Vertical asymptotes:

X^2 - 1 = 0

X^2 = 1

X = 1

X = -1

f’(x) = [2x(x^2 -1) - 2x(x^2 +1)]/(x^2 -1)^2 = 2x(x^2 -1 -x^2 -1)/(x^2 -1)^2 = -4x/(x^2 -1)^2

-4x = 0

X = 0 is a critical point

X<0 f(x) is increasing

x>0 f(x) is decreasing

x = 0 is the maximum of the function

f”(x) =[ -4(x^2 -1)^2 - 2(x^2 -1) * -4x]/(x^2 -1)^4 = 4(-x^2 +x +1)/(x^2 -1)^3

x = +-1 not in the domain

-x^2 +x + 1 = 0

(-1 +-sqrt(1 - 4 * -1 + 1)/-2

X = -1 + sqrt(5)/-2 = -0.62

X = 1.62

confidence rating #$&*:

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Given Solution:

`a First we look for zeros and intercepts:

The numerator is never zero, being the sum of the positive number 1 and the nonnegative quantity x^2. So the function has no zeros, i.e., never crosses the x axis.

When x=0 we have y = (0^2+1)/(0^2-1) = -1 so the y intercept is (0,-1).

Next we analyze the derivative to see if we can find relative maxima and minima:

The derivative is - 4x/(x^2 - 1)^2, which has its only zero when x = 0. So (0, -1) is the only critical point.

The second derivative is 4(3x^2 + 1)/(x^2 - 1)^3, which is negative when x = 0. So the critical point gives us a maximum.

We analyze the second derivative to determine concavity:

The second derivative 4(3x^2 + 1)/(x^2 - 1)^3 has a numerator which is always positive, since x^2 is always positive. The denominator is negative where x^2 - 1 < 0, which occurs between x = -1 and x = 1.

So the 2d derivative is positive on (-infinity, -1) and on (1, infinity), where the graph will be concave up, and negative on (-1, 1), where the graph will be concave down.

Now we look for vertical and horizontal asymptotes:

The denominator of (x^2+1)/(x^2-1) is zero and the numerator isn't when x = +1 and also when x = -1. So we have vertical asymptotes at x = +1 and at x = -1.

}For very large positive x or for very large negative x the +1 in the numerator and the -1 in the denominator are both insignificant and the function value is very close to x^2 / x^2 = 1. The function approaches its horizontal asymptote y = 1 for both large positive x and large negative x.

We finally determine where the function is positive and where negative:

For x < -1 the function cannot change sign, since it is continuous and has no zeros. Testing any x < -1 gives us a positive value. On this interval the function is therefore positive.

The same is true for x > 1.

For -1 < x < 1 the same argument shows that the function is negative**

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Self-critique (if necessary):

I messed up the second derivative… the rest seems fine.

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Self-critique (if necessary):

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Self-critique rating:

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Question: `qQuery problem 1 g 8th edition 3.7.34 7th edition 3.7.32 sketch (x^2+1)/(x^2-1)

Note: The problem in the text might be (x^2+1)/(x^2-2). If so the solution given below can be easily adapted to that function.

Describe your graph, including a description of any intercepts, extrema, asymptotes and concavity.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

X intercepts:

X^2 +1 = 0

X^2 = -1 there are no x intercepts

Y intercepts: y = -1

Horizontal asymptote is y = 1

Vertical asymptotes:

X^2 - 1 = 0

X^2 = 1

X = 1

X = -1

f’(x) = [2x(x^2 -1) - 2x(x^2 +1)]/(x^2 -1)^2 = 2x(x^2 -1 -x^2 -1)/(x^2 -1)^2 = -4x/(x^2 -1)^2

-4x = 0

X = 0 is a critical point

X<0 f(x) is increasing

x>0 f(x) is decreasing

x = 0 is the maximum of the function

f”(x) =[ -4(x^2 -1)^2 - 2(x^2 -1) * -4x]/(x^2 -1)^4 = 4(-x^2 +x +1)/(x^2 -1)^3

x = +-1 not in the domain

-x^2 +x + 1 = 0

(-1 +-sqrt(1 - 4 * -1 + 1)/-2

X = -1 + sqrt(5)/-2 = -0.62

X = 1.62

confidence rating #$&*:

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Given Solution:

`a First we look for zeros and intercepts:

The numerator is never zero, being the sum of the positive number 1 and the nonnegative quantity x^2. So the function has no zeros, i.e., never crosses the x axis.

When x=0 we have y = (0^2+1)/(0^2-1) = -1 so the y intercept is (0,-1).

Next we analyze the derivative to see if we can find relative maxima and minima:

The derivative is - 4x/(x^2 - 1)^2, which has its only zero when x = 0. So (0, -1) is the only critical point.

The second derivative is 4(3x^2 + 1)/(x^2 - 1)^3, which is negative when x = 0. So the critical point gives us a maximum.

We analyze the second derivative to determine concavity:

The second derivative 4(3x^2 + 1)/(x^2 - 1)^3 has a numerator which is always positive, since x^2 is always positive. The denominator is negative where x^2 - 1 < 0, which occurs between x = -1 and x = 1.

So the 2d derivative is positive on (-infinity, -1) and on (1, infinity), where the graph will be concave up, and negative on (-1, 1), where the graph will be concave down.

Now we look for vertical and horizontal asymptotes:

The denominator of (x^2+1)/(x^2-1) is zero and the numerator isn't when x = +1 and also when x = -1. So we have vertical asymptotes at x = +1 and at x = -1.

}For very large positive x or for very large negative x the +1 in the numerator and the -1 in the denominator are both insignificant and the function value is very close to x^2 / x^2 = 1. The function approaches its horizontal asymptote y = 1 for both large positive x and large negative x.

We finally determine where the function is positive and where negative:

For x < -1 the function cannot change sign, since it is continuous and has no zeros. Testing any x < -1 gives us a positive value. On this interval the function is therefore positive.

The same is true for x > 1.

For -1 < x < 1 the same argument shows that the function is negative**

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Self-critique (if necessary):

I messed up the second derivative… the rest seems fine.

"

Self-critique (if necessary):

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Self-critique rating:

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&#Very good responses. Let me know if you have questions. &#