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brilliants [131]
2 years ago
7

Classwork/Homework:

Mathematics
1 answer:
REY [17]2 years ago
3 0
An object's dimensions are: length of 25cm, wight of 10cm, and height of 15cm. calculate the object's density if its mass is 3000g
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CALCULUS - Find the values of in the interval (0,2pi) where the tangent line to the graph of y = sinxcosx is
Rufina [12.5K]

Answer:

\{\frac{\pi}{4}, \frac{3\pi}{4},\frac{5\pi}{4},\frac{7\pi}{4}\}

Step-by-step explanation:

We want to find the values between the interval (0, 2π) where the tangent line to the graph of y=sin(x)cos(x) is horizontal.

Since the tangent line is horizontal, this means that our derivative at those points are 0.

So, first, let's find the derivative of our function.

y=\sin(x)\cos(x)

Take the derivative of both sides with respect to x:

\frac{d}{dx}[y]=\frac{d}{dx}[\sin(x)\cos(x)]

We need to use the product rule:

(uv)'=u'v+uv'

So, differentiate:

y'=\frac{d}{dx}[\sin(x)]\cos(x)+\sin(x)\frac{d}{dx}[\cos(x)]

Evaluate:

y'=(\cos(x))(\cos(x))+\sin(x)(-\sin(x))

Simplify:

y'=\cos^2(x)-\sin^2(x)

Since our tangent line is horizontal, the slope is 0. So, substitute 0 for y':

0=\cos^2(x)-\sin^2(x)

Now, let's solve for x. First, we can use the difference of two squares to obtain:

0=(\cos(x)-\sin(x))(\cos(x)+\sin(x))

Zero Product Property:

0=\cos(x)-\sin(x)\text{ or } 0=\cos(x)+\sin(x)

Solve for each case.

Case 1:

0=\cos(x)-\sin(x)

Add sin(x) to both sides:

\cos(x)=\sin(x)

To solve this, we can use the unit circle.

Recall at what points cosine equals sine.

This only happens twice: at π/4 (45°) and at 5π/4 (225°).

At both of these points, both cosine and sine equals √2/2 and -√2/2.

And between the intervals 0 and 2π, these are the only two times that happens.

Case II:

We have:

0=\cos(x)+\sin(x)

Subtract sine from both sides:

\cos(x)=-\sin(x)

Again, we can use the unit circle. Recall when cosine is the opposite of sine.

Like the previous one, this also happens at the 45°. However, this times, it happens at 3π/4 and 7π/4.

At 3π/4, cosine is -√2/2, and sine is √2/2. If we divide by a negative, we will see that cos(x)=-sin(x).

At 7π/4, cosine is √2/2, and sine is -√2/2, thus making our equation true.

Therefore, our solution set is:

\{\frac{\pi}{4}, \frac{3\pi}{4},\frac{5\pi}{4},\frac{7\pi}{4}\}

And we're done!

Edit: Small Mistake :)

5 0
2 years ago
Electrical Resistance The electrical resistance R of a wire varies inversely with the square of its diameter d. If a 25-foot wir
kati45 [8]

Answer:

The answer is below

Step-by-step explanation:

The electrical resistance R of a wire varies inversely with the square of its diameter d:

R ∝ 1 / d

Let k represent the constant of proportionality, hence we can represent the statement as an equation:

R=\frac{k}{d^2}\\ \\when\ R=0.5 \Omega,d=2\ mm\\\\0.5=\frac{k}{2^2} \\\\k=2\\\\The\ equation\ becomes :\\\\R=\frac{2}{d^2}\\ \\When\ d=3, we\ find\ R\\\\R=\frac{2}{3^2} \\\\R=0.22\ \Omega

8 0
3 years ago
The product of twice a number and four is the same as the difference of seven times the number and 1/6
mamaluj [8]

Answer:

x=-1/6

Step-by-step explanation:

In the attached file

5 0
3 years ago
Paige works at a department store and has a commission rate of 3 percent. Her sales for the quarter were $4,596. If she earns $5
ivanzaharov [21]

Here's what the equation would look like:

30(50) + 0.03(4596)

1500 + 137.88

Paige earned $1637.88.

8 0
2 years ago
The height of a tree was 8 feet. After a year the trees height increased by 25% draw a diagram to find the new height of the tre
Over [174]
Wait you can draw on here?

8 0
2 years ago
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