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2 years ago

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The tangent of the curve f(t) =2+nt+mt^2 at point (1,1/2) is parallel to the normal g(t)= t^2+6t+13(-2,2), calculate the values

m and n.

1 answer:

Anettt [7]2 years ago

7 0

Step-by-step explanation:

The slope of g(t) as (-2,2) is given by

$g'(t)=2t+6$

$g'(-2)=2(-2)+6=2$

Since the normal is parallel to the tangent of f(t) at (1, 1/2),

The tangent line of f(t) has a slope of

$\dfrac{d}{dx}f(t)=n+2mt=2$

At point (1, 1/2), we know

$f'(1)=n+2m=2$

$f(1)=2+n(1)+m(1)^2=2+n+m=\frac12$

Solving,

$n=5, m=-\frac72$

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3 years ago

What is another way to group the factors (3x2)x5

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3*(2*5)
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3 years ago

How to know if a function is periodic without graphing it ?

zhenek [66]

A function $f(t)$ is periodic if there is some constant $k$ such that $f(t+k)=f(k)$ for all $t$ in the domain of $f(t)$ . Then $k$ is the "period" of $f(t)$ .

Example:

If $f(x)=\sin x$ , then we have $\sin(x+2\pi)=\sin x\cos2\pi+\cos x\sin2\pi=\sin x$ , and so $\sin x$ is periodic with period $2\pi$ .

It gets a bit more complicated for a function like yours. We're looking for $k$ such that

$\pi\sin\left(\dfrac\pi2(t+k)\right)+1.8\cos\left(\dfrac{7\pi}5(t+k)\right)=\pi\sin\dfrac{\pi t}2+1.8\cos\dfrac{7\pi t}5$

Expanding on the left, you have

$\pi\sin\dfrac{\pi t}2\cos\dfrac{k\pi}2+\pi\cos\dfrac{\pi t}2\sin\dfrac{k\pi}2$

and

$1.8\cos\dfrac{7\pi t}5\cos\dfrac{7k\pi}5-1.8\sin\dfrac{7\pi t}5\sin\dfrac{7k\pi}5$

It follows that the following must be satisfied:

$\begin{cases}\cos\dfrac{k\pi}2=1\\\\\sin\dfrac{k\pi}2=0\\\\\cos\dfrac{7k\pi}5=1\\\\\sin\dfrac{7k\pi}5=0\end{cases}$

The first two equations are satisfied whenever $k\in\{0,\pm4,\pm8,\ldots\}$ , or more generally, when $k=4n$ and $n\in\mathbb Z$ (i.e. any multiple of 4).

The second two are satisfied whenever $k\in\left\{0,\pm\dfrac{10}7,\pm\dfrac{20}7,\ldots\right\}$ , and more generally when $k=\dfrac{10n}7$ with $n\in\mathbb Z$ (any multiple of 10/7).

It then follows that all four equations will be satisfied whenever the two sets above intersect. This happens when $k$ is any common multiple of 4 and 10/7. The least positive one would be 20, which means the period for your function is 20.

Let's verify:

$\sin\left(\dfrac\pi2(t+20)\right)=\sin\dfrac{\pi t}2\underbrace{\cos10\pi}_1+\cos\dfrac{\pi t}2\underbrace{\sin10\pi}_0=\sin\dfrac{\pi t}2$

$\cos\left(\dfrac{7\pi}5(t+20)\right)=\cos\dfrac{7\pi t}5\underbrace{\cos28\pi}_1-\sin\dfrac{7\pi t}5\underbrace{\sin28\pi}_0=\cos\dfrac{7\pi t}5$

More generally, it can be shown that

$f(t)=\displaystyle\sum_{i=1}^n(a_i\sin(b_it)+c_i\cos(d_it))$

is periodic with period $\mbox{lcm}(b_1,\ldots,b_n,d_1,\ldots,d_n)$ .

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3 years ago

If two distinct planes intersect , then their intersection is a line . Which geometry term does the statement represent

Rom4ik [11]

Answer:

POSTULATE

Step-by-step explanation:

<em>A postulate is a statement that is assumed true without proof. A theorem is a true statement that can be proven </em>

<em> </em>

<em> </em>

<em> </em>

<em>In geometry, the theorem is considered valid or correct with which its assertion is practically not refuted by almost anyone, whether it is really true or not. </em>

<em> </em>

<em> </em>

<em> </em>

<em>This, in practice, implies that all students of the subject share the same criteria on the theory. Thus, the postulates, in short, have the following characteristics: </em>

<em> </em>

<em> They are presumed true by most of the scholars in the field. </em>

<em> </em>

<em>Its contradiction goes against the very essence of the subject. </em>

<em> </em>

<em>Postulate 6: If two planes intersect, then their intersection is a line.</em>

<em />

<em>See Attachment for Details ^^</em>

<em>The image attached represents the statement given in the question (If two distinct planes intersect, then their intersection is a line)</em>

<em>Therefore</em><em>, </em>

<em> </em>

<em>The Geometry term that represents the statement above is POSTULATE</em>

5 0

2 years ago