Find angle x.<br> Try not to tell me the wrong answers...<br> A. 144<br> B. 70<br> C. 36<br> D. 90

<img src="https://tex.z-dn.net/?f=%5Clim_%7Bx%5Cto%20%5C%200%7D%20%5Cfrac%7B%5Csqrt%7Bcos2x%7D-%5Csqrt%5B3%5D%7Bcos3x%7D%20%7D%7

salantis [7]

Answer:

$\displaystyle \lim_{x \to 0} \frac{\sqrt{cos(2x)} - \sqrt[3]{cos(3x)}}{sin(x^2)} = \frac{1}{2}$

General Formulas and Concepts:

<u>Calculus</u>

Limits

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

L'Hopital's Rule

Differentiation

Derivatives
Derivative Notation

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:

We are given the limit:

$\displaystyle \lim_{x \to 0} \frac{\sqrt{cos(2x)} - \sqrt[3]{cos(3x)}}{sin(x^2)}$

When we directly plug in <em>x</em> = 0, we see that we would have an indeterminate form:

$\displaystyle \lim_{x \to 0} \frac{\sqrt{cos(2x)} - \sqrt[3]{cos(3x)}}{sin(x^2)} = \frac{0}{0}$

This tells us we need to use L'Hoptial's Rule. Let's differentiate the limit:

$\displaystyle \lim_{x \to 0} \frac{\sqrt{cos(2x)} - \sqrt[3]{cos(3x)}}{sin(x^2)} = \displaystyle \lim_{x \to 0} \frac{\frac{-sin(2x)}{\sqrt{cos(2x)}} + \frac{sin(3x)}{[cos(3x)]^{\frac{2}{3}}}}{2xcos(x^2)}$

Plugging in <em>x</em> = 0 again, we would get:

$\displaystyle \lim_{x \to 0} \frac{\frac{-sin(2x)}{\sqrt{cos(2x)}} + \frac{sin(3x)}{[cos(3x)]^{\frac{2}{3}}}}{2xcos(x^2)} = \frac{0}{0}$

Since we reached another indeterminate form, let's apply L'Hoptial's Rule again:

$\displaystyle \lim_{x \to 0} \frac{\frac{-sin(2x)}{\sqrt{cos(2x)}} + \frac{sin(3x)}{[cos(3x)]^{\frac{2}{3}}}}{2xcos(x^2)} = \lim_{x \to 0} \frac{\frac{-[cos^2(2x) + 1]}{[cos(2x)]^{\frac{2}{3}}} + \frac{cos^2(3x) + 2}{[cos(3x)]^{\frac{5}{3}}}}{2cos(x^2) - 4x^2sin(x^2)}$

Substitute in <em>x</em> = 0 once more:

$\displaystyle \lim_{x \to 0} \frac{\frac{-[cos^2(2x) + 1]}{[cos(2x)]^{\frac{2}{3}}} + \frac{cos^2(3x) + 2}{[cos(3x)]^{\frac{5}{3}}}}{2cos(x^2) - 4x^2sin(x^2)} = \frac{1}{2}$

And we have our final answer.

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

Unit: Limits

6 0

2 years ago

a carpebter has a board 2 meter needs to cut it in 8 equal pieces how many centerimeters long will each piece be

likoan [24]

2 centerimeters each

6 0

3 years ago

Which value of x is a solution of x < -2?<br> A.<br> -3<br> B. -2<br> C.<br> -1<br> D. 0

Viefleur [7K]

Answer:

A

Step-by-step explanation:

-3<-2

7 0

3 years ago

Which of the pairs of ratios are equivalent? A) 12: 24, 50 :100 B) 16 to 3,27 to 5 C) 22/1, 68/3 D) 3/7, 17/35

RoseWind [281]

Answer:

A

Step-by-step explanation:

equivalent because each are 1:2

3 0

2 years ago

Consider the following data on the estimated cost (in millions of dollars) resulting from traffic congestion for different urban

IrinaK [193]

The mean is $5.769 million and the standard deviation is $3.905 million.

The mean is the average of a data set.

<h3>What is the standard deviation?</h3>

The standard deviation shows the measure of the dispersion from the mean.

<h3>Data and Calculations:</h3>

Urban Area Total Cost Squared Difference

(millions of dollars) (x - u)²

New York 16 104.65

Los Angeles 13 52.27

Chicago 7 1.51

Washington, D.C. 5 0.59

Houston 5 0.59

Dallas, Fort Worth 4 3.13

Detroit 4 3.13

Miami 4 3.13

Phoenix 4 3.13

Philadelphia 4 3.13

San Francisco 3 7.67

Boston 3 7.67

Atlanta 3 7.67

13 Total 75 198.27

Mean = 5.769 (75/13) 15.251 (198.27/13)

Standard Deviation = 3.905 (square root of 15.251)

<h3>Question Completion:</h3>

Urban Area Total Cost

(millions of dollars)

New York 16

Los Angeles 13

Chicago 7

Washington, D.C. 5

Houston 5

Dallas, Fort Worth 4

Detroit 4

Miami 4

Phoenix 4

Philadelphia 4

San Francisco 3

Boston 3

Atlanta 3

Calculate the mean and standard deviation for this data set (in millions of dollars). (Round your answers to three decimal places.) mean $ million standard deviation $ million

Thus, the mean is $5.769 million and the standard deviation is $3.905 million.

Learn more about calculating the mean and standard deviation at brainly.com/question/1580527

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6 0

1 year ago

Find angle x. Try not to tell me the wrong answers... A. 144 B. 70 C. 36 D. 90