All categories

1 year ago

If f(x) = f(x)g(x), where f and g have derivatives of all orders, show that f'' = f ''g + 2f 'g' + fg''.

1 answer:

5 0

Differentiating once, we have

$f'(x)=f'(x)g(x)+f(x)g'(x)$

Differentiating again,

$f''(x)=f''(x)g(x)+f'(x)g'(x)+f'(x)g'(x)+f(x)g''(x)$
$f''(x)=f''(x)g(x)+2f'(x)g'(x)+f(x)g''(x)$

as needed.

You might be interested in

An energy drink container in the shape of a right circular cylinder must have a volume of 19 fluid ounces (1 fluid ounce is appr

omeli [17]

Answer:

the dimension that will minimize cost are radius of 2.79 inch and height of cylinder of 1.40 inch.

Step-by-step explanation:

volume of a circular cylinder = πr²h

volume of 19 fluid ounces (1 fluid ounce is approximately 1.80469 cubic inches) = 19* 1.80469 =34.28911 in³

The cost per square inch of constructing the top and bottom is twice the cost of constructing the lateral side, means

surface area of top + bottom = surface area of side

area of circle(top + bottom) = πr²+πr² = 2πr²

area of the side = 2πrh

the cost of top and bottom = twice the cost of side

2πr² = 2( 2πrh) = 4πrh

divide both side by 2πr

r= 4πrh/ 2πr =2h

r = 2h

for volume = πr²h = 34.28911 in³

and r = 2h

π(2h)²h =34.28911 in³

4πh³ = 34.28911 in³

h³ = 34.28911 /(4*3.142) = 2.7283 in³

h = cube root of 2.7283 in³ = 1.397 inch

r = 2* 1.76 = 2.794 inch

the dimension that will minimize cost are radius of 2.79 inch and height of cylinder of 1.40 inch.

5 0

2 years ago

Ethan bought 4 packages of pencils. After he gave 8 pencils to his friends, he had 40 pencils left over. How many pencils were i

swat32

Answer:

12 pencils in each package.

Step-by-step explanation:

Given:

Ethan bought 4 packages of pencils.

No of Packages = 4

No of pencils given to friend = 8 pencils

No of pencils left over =40 pencils

∴Total Number of Pencils = No of pencils given to friend + No of pencils left over = $40+8 =48$

Total Number of pencils in each package = $\frac{\textrm{Total Number of Pencils}}{\textrm{No Of Packages}}= \frac{48}{4}= 12 \ pencils$

8 0

2 years ago

Phoebe surveyed teens and adults to find the average number of hours of sleep they get during the week. Her results are shown be

Thepotemich [5.8K]

Answer:

its the second option

Step-by-step explanation:

i did the test and got 100

8 0

2 years ago

Read 2 more answers

Chelsea has 11 times as many art brushes as Monique. If they have 60 art brushes altogether, how many brushes does Chelsea have?

notsponge [240]

Let Monique's number of art brushes be m.
Then, since Chelsea has <span>11 times as many art brushes as Monique, Chelsea has 11m brushes.

Together they have 60 art brushes, so we write the equation:

m+11m=60.

This means that 12m=60, dividing both sides of the equation by 12 we get m=5.

Chelsea has 11m=11*5=55 brushes.

Answer: 55</span>

7 0

2 years ago

The random variable X is normally distributed with mean 82 and standard deviation 7.4. Find the value of q such that P(82 − q &l

mezya [45]

P(82 - q < x < 82 + q) = 0.44
P(x < 82 + q) - P(82 - q) = 0.44
P(z < (82 + q - 82)/7.4 - P(z < (82 - q - 82)/7.4) = 0.44
P(z < q/7.4) - P(z < -q/7.4) = 0.44
P(z < q/7.4) - (1 - P(z < q/7.4) = 0.44
P(z < q/7.4) - 1 + P(z < q/7.4) = 0.44
2P(z < q/7.4) - 1 = 0.44
2P(z < q/7.4) = 1.44
P(z < q/7.4) = 0.72
P(z < q/7.4) = P(z < 0.583)
q/7.4 = 0.583
q = 0.583 x 7.4 = 4.31

8 0

1 year ago