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2 years ago

Determine the sum of the first 1000 Natural numbers

Mathematics

1 answer:

levacccp [35]2 years ago

8 0

Answer:

Numbers Sum

1-10 55

1-100 5050

1-1000 500500

Step-by-step explanation:

55+ 5050=

500500

500500 is a sum of number series from 1 to 1000 by applying the values of input parameters in the formula. Natural numbers or positive integers from 1 to 1000 by applying arithmetic progression. (to find it)

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3 years ago

Find the tangent line approximation for 10+x−−−−−√ near x=0. Do not approximate any of the values in your formula when entering

Svetllana [295]

Answer:

$L(x)=\sqrt{10}+\frac{\sqrt{10}}{20}x$

Step-by-step explanation:

We are asked to find the tangent line approximation for $f(x)=\sqrt{10+x}$ near $x=0$ .

We will use linear approximation formula for a tangent line $L(x)$ of a function $f(x)$ at $x=a$ to solve our given problem.

$L(x)=f(a)+f'(a)(x-a)$

Let us find value of function at $x=0$ as:

$f(0)=\sqrt{10+x}=\sqrt{10+0}=\sqrt{10}$

Now, we will find derivative of given function as:

$f(x)=\sqrt{10+x}=(10+x)^{\frac{1}{2}}$

$f'(x)=\frac{d}{dx}((10+x)^{\frac{1}{2}})\cdot \frac{d}{dx}(10+x)$

$f'(x)=\frac{1}{2}(10+x)^{-\frac{1}{2}}\cdot 1$

$f'(x)=\frac{1}{2\sqrt{10+x}}$

Let us find derivative at $x=0$

$f'(0)=\frac{1}{2\sqrt{10+0}}=\frac{1}{2\sqrt{10}}$

Upon substituting our given values in linear approximation formula, we will get:

$L(x)=\sqrt{10}+\frac{1}{2\sqrt{10}}(x-0)$

$L(x)=\sqrt{10}+\frac{1}{2\sqrt{10}}x-0$

$L(x)=\sqrt{10}+\frac{\sqrt{10}}{20}x$

Therefore, our required tangent line for approximation would be $L(x)=\sqrt{10}+\frac{\sqrt{10}}{20}x$ .

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3 years ago

Find the midpoint of the segment with the given endpoints.<br> (3,- 9) and (4, -8)

frez [133]

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$\boxed{\sf (x,y)=\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)}$

$\\ \sf\longmapsto \left(\dfrac{3+4}{2},\dfrac{-9+8}{2}\right)$

$\\ \sf\longmapsto \left(\dfrac{7}{2},\dfrac{-1}{2}\right)$

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3 years ago