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Firlakuza [10]
2 years ago
10

find the values for c and d that make the function differentiable:f(x) = 5x^2 + c if x = or < than 3dx^2 - 3 if x > 3

Mathematics
1 answer:
daser333 [38]2 years ago
4 0

9514 1404 393

Answer:

  c = -3; d = 5

Step-by-step explanation:

The function is differentiable if it is continuous and the derivative is continuous.

This function will be continuous if the limit as x approaches 3 from either side is the same. From the left, the limit is ...

  5(3^2) +c = 45 +c

From the right, the limit is ...

  d(3^2) -3 = 9d -3

__

The derivative will be continuous if the limit as x approaches 3 from either side is the same. From the left, the limit is ...

  f'(x) = 10x

  = 10(3) = 30

From the right, the limit is ...

  f'(x) = 2dx

  = 2d(3) = 6d

__

This gives us a pair of simultaneous equations in 'c' and 'd':

  45 +c = 9d -3

  6d = 30

The latter tells us d = 5. Then the former tells us ...

  45 +c = 9(5) -3   ⇒   c = -3

The function is differentiable if c = -3 and d = 5.

_____

<em>Additional comment</em>

With these values, the function becomes ...

  f(x)=\begin{cases}5x^2-3&\text{if }x\le 3\\5x^2-3&\text{if }x>3\end{cases}

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A giant scoop, operated by a crane, is in the shape of a hemisphere of radius 21 inches. The scoop is used to transfer molten st
eduard

Answer:

31.5 in

Step-by-step explanation:

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    (where r is the radius)
  • Volume of a cylinder = \pir²h
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  • radius r = (1/2) diameter

First, find the volume of the scoop using the volume of a hemisphere formula with r = 21:

Volume = (2/3)\pi x 21³ = 6174\pi in³

Now equate the found volume of the scoop to the equation of the volume of a cylinder with r = 14, and solve for h:

                                            \pi14²h = 6174\pi

                                            196\pih = 6174\pi

 Divide both sides by \pi:        196h = 6174

Divide both sides by 196:          h = 31.5

Therefore, the height of the molten steel in the storage tank is 31.5 in

3 0
2 years ago
An isosceles triangle has 2 equivalent sides. The perimeter of an isosceles triangle is 84 centimeters. The measure of one of th
Pani-rosa [81]

Well, to find your solution to this problem, I would subtract 84 from 32 = 52. I would divide the number by 2, which equals 26. So, side lengths are 32, 26, and 26. 

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Data consistent with summary quantities in the article "Effects of Fast-Food Consumption on Energy Intake and Diet Quality Among
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Answer:

a) X[bar]₁=  1839.20 cal

b) X[bar]₂= 1779.07 cal

c) S₁= 386.35 cal

Step-by-step explanation:

Hello!

You have two independent samples,

Sample 1: n₁= 15 children that did not eat fast food.

Sample 2: n₂= 15 children that ate fast food.

The study variables are:

X₁: Calorie consumption of a kid that does not eat fast food in one day.

X₂: Calorie consumprion of a kid that eats fast food in one day.

a)

The point estimate of the population mean is the sample mean

X[bar]₁= (∑X₁/n₁) = (27588/15)= 1839.20 cal

b)

X[bar]₂= (∑X₂/n₂)= (26686/15)= 1779.07 cal

c)

To calculate the sample standard deiation, you have to calculate the sample variance first:

S₁²= \frac{1}{n_1-1}[∑X₁² - (( ∑X₁)²/n₁)]= \frac{1}{14} + [52829538 - (\frac{27588^{2} }{15}) = 149263.4571 cal²

S₁= 386.35 cal

I hope it helps!

7 0
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The percent decreased by 65%
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