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Gre4nikov [31]
3 years ago
7

Find the Horizontal Tangent for x^3/3+3x^2-16x+9

Mathematics
1 answer:
bazaltina [42]3 years ago
3 0

Answer:

The two horiz. tang. lines here are y = -3 and y = 192.

Step-by-step explanation:

Remember that the slope of a tangent line to the graph of a function is given by the derivative of that function.  Thus, we find f '(x):

f '(x) = x^2 + 6x - 16.  This is the formula for the slope.  We set this = to 0 and determine for which x values the tangent line is horizontal:

f '(x) = x^2 + 6x - 16 = 0.  Use the quadratic formula to determine the roots here:  a = 1; b = 6 and c = -16:  the discriminant is b^2-4ac, or 36-4(1)(-16), which has the value 100; thus, the roots are:

      -6 plus or minus √100

x = ----------------------------------- = 2 and -8.

                         2

Evaluating y = x^3/3+3x^2-16x+9 at x = 2 results in y = -3.  So one point of tangency is (2, -3).  Remembering that the tangent lines in this problem are horizontal, we need only the y-coefficient of (2, -3) to represent this first tangent line:  it is y = -3.

Similarly, find the y-coeff. of the other tangent line, which is tangent to the curve at x = -8.  The value of x^3/3+3x^2-16x+9 at x = -8 is  192, and so the equation of the 2nd tangent line is y=192 (the slope is zero).


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The density of a certain material is such that it weighs 71 pounds for every 2.5 gallons of volume. Express this density in tons
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Step-by-step explanation:

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Find the interior angle sum for each polygon. Round your answer to the nearest tenth if needed
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Answer:

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3) x = -1

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Step-by-step explanation:

The formula for finding the sum of interior angles in a polygon is the following,

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Where (S) represents the sum of interior angles, and (n) represents the number of sides.

1)

The given polygon is a hexagon, meaning it has (6) sides. Substitute the number of sides into the formula and solve for the sum of interior angles.

S=180(n-2)

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2)

This polygon is a pentagon, meaning it has (5) sides. Substitute the number of sides into the formula and solve for the sum of interior angles.

S=180(n-2)

S=180(5-2)\\\\S=180(3)\\\\S=540

3)

The sum of interior angles in a triangle is (180) degrees. One can see that one of the measures of an angle is (90) degrees (indicated by the box around the angle). One can form an equation by adding up the expressions for the angle measures and setting it equal to (180) degrees.

(55) + (x + 36) + (90) = 180

Simplify,

181 + x = 180

Inverse operations,

181 + x = 180

x = -1

4)

The remote angles theorem states that when one extends one of the sides of a triangle, the angle formed between the side and the extension is equal to the sum of the two non-adjacent interior angles in the triangle. Based on this theorem, one can form an equation and solve for the unknown.

<F + <G = <FEZ

Substitute,

(4x) + (-5 + 7x) = 105

Simplify,

4x - 5 + 7x = 105

11x - 5 = 105

Inverse operations,

11x - 5 = 105

11x = 110

x = 10

6 0
2 years ago
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