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olya-2409 [2.1K]
2 years ago
10

Fundamental theorem of calculus

%20%5C%2C%20dt" id="TexFormula1" title="g(s)=\int\limits^s_6 {(t-t^4)^6} \, dt" alt="g(s)=\int\limits^s_6 {(t-t^4)^6} \, dt" align="absmiddle" class="latex-formula">
Mathematics
1 answer:
mr_godi [17]2 years ago
3 0

Answer:

\displaystyle g'(s) = (s-s^4)^6

Step-by-step explanation:

The Fundamental Theorem of Calculus states that:
\displaystyle \frac{d}{dx}\left[ \int_a^x f(t)\, dt  \right] = f(x)

Where <em>a</em> is some constant.

We can let:
\displaystyle g(t) = (t-t^4)^6

By substitution:

\displaystyle g(s) = \int_6^s g(t)\, dt

Taking the derivative of both sides results in:
\displaystyle g'(s) = \frac{d}{ds}\left[ \int_6^s g(t)\, dt\right]

Hence, by the Fundamental Theorem:

\displaystyle \begin{aligned} g'(s) & = g(s) \\ \\  & = (s-s^4)^6\end{aligned}

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In the rectangle below, EI = 2x+6, FI = 5x-6, and m ZIFE = 49º.
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Answer:

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Step-by-step explanation:

6 0
3 years ago
A bowl of fruit is on the table. It contains 5 apples, 2 oranges, and 2 bananas. Christian and Aaron come home from school and r
Sholpan [36]

The probability that both grab oranges would be = \frac{1}{36}

Step-by-step explanation:

Given,

A bowl contains,

Apples = a = 5

Oranges = o = 2

bananas = b = 2

Total fruits in bowl = x = a + o + b = 5 + 2 + 2 = 9

Now, Christian and Aaron come home from school and randomly grab one fruit each.

The probability of first selecting orange would be = \frac{o}{b} = \frac{2}{9}

Now, the oranges left would be 2 - 1 = 1

and total fruits would be = 9 - 1 = 8

Hence, the probability of selecting second orange would be = \frac{1}{8}

Therefore, the probability that both grab oranges would be = \frac{1}{8} * \frac{2}{9} = \frac{1}{36}

7 0
3 years ago
What is the equation of the quadratic graph with a focus of (1, 1) and a directrix of y = −1? PLs HELP!!
Delicious77 [7]
ANSWER
f(x)= \frac{1}{4} {(x - 1)}^{2}

EXPLANATION

Since the directrix is
y = - 1

the axis of symmetry of the parabola is parallel to the y-axis.

Again, the focus being,

(1,1)

also means that the parabola will open upwards.

The equation of parabola with such properties is given by,

{(x - h)}^{2} = 4p(y - k)
where
(h,k)

is the vertex of the parabola.

The directrix and the axis of symmetry of the parabola will intersect at
(1, - 1)

The vertex is the midpoint of the focus and the point of intersection of the axis of the parabola and the directrix.

This implies that,
h = \frac{1 + 1}{2} = 1

and
k = \frac{ - 1 + 1}{2} = 0

The equation of the parabola now becomes,

(x - 1) ^{2} = 4p(y - 0)

|p| = 1

Thus, the distance between the vertex and the directrix.

This means that,

p = - 1 \: or \: 1

Since the parabola opens up, we choose
p = 1
Our equation now becomes,

{(x - 1)}^{2} = 4(1)(y - 0)

This simplifies to
{(x - 1)}^{2} = 4y

or

y = \frac{1}{4} {(x - 1)}^{2}

This is the same as,

f(x)= \frac{1}{4} {(x - 1)}^{2}

The correct answer is D .
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3 years ago
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brilliants [131]
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3 years ago
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Alborosie

Answer:

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Step-by-step explanation:

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7 0
3 years ago
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