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3 years ago

14

Jade had the following system of equations to help her with a physics problem, but she only needed to know what the sum of x and

y is.
2x+3y=12
2x-y=4
What is the value of x+y?

1 answer:

pav-90 [236]3 years ago

8 0

From the second:

2x-y=4

x=(4+y)/2 and applying this to first:

4+y+3y=12

4y+4=12

4y=8

y=2 and since x=(4+y)/2, x=3 so

x+y=2+3=5

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2 years ago

What is Limit of StartFraction StartRoot x + 1 EndRoot minus 2 Over x minus 3 EndFraction as x approaches 3?

scoray [572]

Answer:

<u /> $\displaystyle \lim_{x \to 3} \frac{\sqrt{x + 1} - 2}{x - 3} = \boxed{ \frac{1}{4} }$

General Formulas and Concepts:

<u>Calculus</u>

Limits

Limit Rule [Variable Direct Substitution]:
$\displaystyle \lim_{x \to c} x = c$

Special Limit Rule [L’Hopital’s Rule]:
$\displaystyle \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$

Differentiation

Derivatives
Derivative Notation

Derivative Property [Addition/Subtraction]:
$\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]$
Derivative Rule [Basic Power Rule]:

f(x) = cxⁿ
f’(x) = c·nxⁿ⁻¹

Derivative Rule [Chain Rule]:
$\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)$

Step-by-step explanation:

<u>Step 1: Define</u>

<em>Identify given limit</em>.

$\displaystyle \lim_{x \to 3} \frac{\sqrt{x + 1} - 2}{x - 3}$

<u>Step 2: Find Limit</u>

Let's start out by <em>directly</em> evaluating the limit:

[Limit] Apply Limit Rule [Variable Direct Substitution]:
$\displaystyle \lim_{x \to 3} \frac{\sqrt{x + 1} - 2}{x - 3} = \frac{\sqrt{3 + 1} - 2}{3 - 3}$
Evaluate:
$\displaystyle \begin{aligned}\lim_{x \to 3} \frac{\sqrt{x + 1} - 2}{x - 3} & = \frac{\sqrt{3 + 1} - 2}{3 - 3} \\& = \frac{0}{0} \leftarrow \\\end{aligned}$

When we do evaluate the limit directly, we end up with an indeterminant form. We can now use L' Hopital's Rule to simply the limit:

[Limit] Apply Limit Rule [L' Hopital's Rule]:
$\displaystyle \begin{aligned}\lim_{x \to 3} \frac{\sqrt{x + 1} - 2}{x - 3} & = \lim_{x \to 3} \frac{(\sqrt{x + 1} - 2)'}{(x - 3)'} \\\end{aligned}$
[Limit] Differentiate [Derivative Rules and Properties]:
$\displaystyle \begin{aligned}\lim_{x \to 3} \frac{\sqrt{x + 1} - 2}{x - 3} & = \lim_{x \to 3} \frac{(\sqrt{x + 1} - 2)'}{(x - 3)'} \\& = \lim_{x \to 3} \frac{1}{2\sqrt{x + 1}} \leftarrow \\\end{aligned}$
[Limit] Apply Limit Rule [Variable Direct Substitution]:
$\displaystyle \begin{aligned}\lim_{x \to 3} \frac{\sqrt{x + 1} - 2}{x - 3} & = \lim_{x \to 3} \frac{(\sqrt{x + 1} - 2)'}{(x - 3)'} \\& = \lim_{x \to 3} \frac{1}{2\sqrt{x + 1}} \\& = \frac{1}{2\sqrt{3 + 1}} \leftarrow \\\end{aligned}$
Evaluate:
$\displaystyle \begin{aligned}\lim_{x \to 3} \frac{\sqrt{x + 1} - 2}{x - 3} & = \lim_{x \to 3} \frac{(\sqrt{x + 1} - 2)'}{(x - 3)'} \\& = \lim_{x \to 3} \frac{1}{2\sqrt{x + 1}} \\& = \frac{1}{2\sqrt{3 + 1}} \\& = \boxed{ \frac{1}{4} } \\\end{aligned}$

∴ we have <em>evaluated</em> the given limit.

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Learn more about limits: brainly.com/question/27807253

Learn more about Calculus: brainly.com/question/27805589

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Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Limits

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1 year ago

What are 3 completed examples for functions?

Lyrx [107]

Example 1

Write y = x2 + 4x + 1 using function notation and evaluate the function at x = 3.

Solution

Given, y = x2 + 4x + 1

By applying function notation, we get

f(x) = x2 + 4x + 1

Evaluation:

Substitute x with 3

f (3) = 32 + 4 × 3 + 1 = 9 + 12 + 1 = 22

Example 2

Evaluate the function f(x) = 3(2x+1) when x = 4.

Solution

Plug x = 4 in the function f(x).

f (4) = 3[2(4) + 1]

f (4) = 3[8 + 1]

f (4) = 3 x 9

f (4) = 27

Example 3

Write the function y = 2x2 + 4x – 3 in function notation and find f (2a + 3).

Solution

y = 2x2 + 4x – 3 ⟹ f (x) = 2x2 + 4x – 3

Substitute x with (2a + 3).

f (2a + 3) = 2(2a + 3)2 + 4(2a + 3) – 3

= 2(4a2 + 12a + 9) + 8a + 12 – 3
= 8a2 + 24a + 18 + 8a + 12 – 3
= 8a2 + 32a + 27

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2 years ago

In the given figure, O is the centre of a circle, TN is a Tangent, AN = 12 cm and OE = 5 cm. Find the length of NE.

sp2606 [1]

Answer:

NE = 8 cm

Step-by-step explanation:

Radius OA = OE = 5 cm

In right triangle OAN, OA² + AN² = ON²

5² + 12² = ON²

ON² = 25 + 144 = 169

so ON = 13 cm

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2 years ago

What is the coordinates of T' if T(-4,2) after a rotation of the triangle 180°about the origin?

guapka [62]

Answer:

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Step-by-step explanation:

Rotation 180° about the origin (same as reflection across the origin) negates both coordinates:

T' = -T = -(-4, 2)

T' = (4, -2)

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2 years ago