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Vikki [24]
2 years ago
6

If f(x) = 2x^2 + 4x + 7, find f'(-5), using the definition of derivative. f'(-5) is the limit as x → -5

Mathematics
1 answer:
rosijanka [135]2 years ago
3 0

Using the limit definition of the derivative, you have

\displaystyle f'(-5) = \lim_{x\to-5} \frac{f(x) - f(-5)}{x - (-5)} = \lim_{x\to-5} \frac{(2x^2+4x+7) - 37}{x + 5}

Simplify the numerator:

(2<em>x</em> ² + 4<em>x</em> + 7) - 37 = 2<em>x</em> ² + 4<em>x</em> - 30

… = 2 (<em>x</em> ² + 2<em>x</em> - 15)

… = 2 (<em>x</em> + 5) (<em>x</em> - 3)

Then

\displaystyle f'(-5) = \lim_{x\to-5}\frac{2(x+5)(x-3)}{x+5} = \lim_{x\to-5} 2(x-3) = \boxed{-16}

• • •

For your second question in the comments, if <em>f(x)</em> = -2<em>x</em> ² + 3<em>x</em> - 7, then by the definition of the derivative, you have

\displaystyle f'(x) = \lim_{h\to0}\frac{f(x+h)-f(x)}h = \lim_{h\to0}\frac{(-2(x+h)^2+3(x+h)-7) - (-2x^2 + 3x - 7)}h

Simplify the numerator:

(-2 (<em>x</em> + <em>h</em>)² + 3 (<em>x</em> + <em>h</em>) - 7) - (-2<em>x</em> ² + 3<em>x</em> - 7)

… = (-2<em>x</em> ² - 4<em>xh</em> - 2<em>h</em> ² + 3<em>x</em> + 3<em>h</em> - 7) - (-2<em>x</em> ² + 3<em>x</em> - 7)

… = -4<em>xh</em> - 2<em>h</em> ² + 3<em>h</em>

Now compute the limit:

\displaystyle f'(x) = \lim_{h\to0}\frac{-4xh-2h^2+3h}h = \lim_{h\to0}(-4x-2h+3) = \boxed{-4x+3}

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Step-by-step explanation:

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= 4 + 12 - 3

= 16 - 3

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Step-by-step explanation:

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Circle C has radius of 10 cm. Each of points B and D is on the midpoint of the radius. Find the area of the shaded region.
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Which set of measurements could represent the three sides of a triangle?
yawa3891 [41]

Answer:

The side lengths of a right triangle is 11cm, 60cm and 61cm, that could be selected from the given measurements.

Step-by-step explanation:

The measurements are,

                  7cm, 11cm, 54cm, 60cm, 61cm, 65cm

Step:1

                 To check the right angle triangle, Pythagorean theorem can be used.

                For a Pythagorean theorem,

                                     ..........................(1)

               The side values are lower than the hypotenuse,

                                                        ...................................(2)

               Where,

                         a,b - side values

                            c - Hypotenuse

               For right angle triangle,  c > a, b

               Alternative : 1

               Take, a = 7cm, b = 11cm

               From eqn (2),

                                                   =  = 13.04

              The above value is not equal to the any one of the values of ( 54cm. 60cm, 61cm, 65cm ), So its not an sides of right triangle.

               Alternative : 2

               Take, a = 7cm, b = 54cm

               From eqn (2),

                                                   =  = 54.45

              The above value is not equal to the any one of the values of (60cm, 61cm, 65cm ), So its not an sides of right triangle.

               Alternative : 3

               Take, a = 7cm, b = 60cm

               From eqn (2),

                                                   =  = 60.406

              The above value is not equal to the any one of the values of (61cm, 65cm ), So its not an sides of right triangle.

               Alternative : 4

               Take, a = 7cm, b = 61cm

               From eqn (2),

                                                   =  = 61.40

              The above value is not equal to the values of (65cm ), So its not an sides of right triangle.

                 Alternative : 5

               Take, a = 11cm, b = 54cm

               From eqn (2),

                                                   =  = 55.1089

              The above value is not equal to the any one of the values of (60cm, 61cm, 65cm ), So its not an sides of right triangle.      

                Alternative : 6

               Take, a = 11cm, b = 60cm

               From eqn (2),

                                                  =  = 61

              The above value is equal to the values of (61cm ), So its an sides of right triangle. The three sides are 11, 60 and 61.

Step:2

            Check for solution,

                                     

                                           

Result:

            The side lengths of a right triangle is 11cm, 60cm and 61cm, that could be selected from the given measurements.                

Step-by-step explanation: The side lengths of a right triangle is 11cm, 60cm and 61cm.

4 0
3 years ago
Type SSS, SAS, ASA, SAA, or HL to<br> describe these triangles.
Yanka [14]

Answer:

SSS

Both these triangles are congruent to each other by SSS congruency

Hope it helps

7 0
3 years ago
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