1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
kicyunya [14]
3 years ago
7

Bsinx%5E%7B2%7D%20%7D" id="TexFormula1" title="\lim_{x\to \ 0} \frac{\sqrt{cos2x}-\sqrt[3]{cos3x} }{sinx^{2} }" alt="\lim_{x\to \ 0} \frac{\sqrt{cos2x}-\sqrt[3]{cos3x} }{sinx^{2} }" align="absmiddle" class="latex-formula">
Mathematics
1 answer:
salantis [7]3 years ago
6 0

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:

  1. f(x) = cxⁿ
  2. 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

You might be interested in
A random sample of 16 students selected from the student body of a large university had an average age of 25 years and a populat
eimsori [14]

Answer:

t=\frac{25-24}{\frac{2}{\sqrt{16}}}=2      

p_v =P(t_{15}>2)=0.0320  

If we compare the p value and the significance level given for example \alpha=0.05 we see that p_v so we can conclude that we reject the null hypothesis, and the true mean is significant higher than 24 years.  

Step-by-step explanation:

1) Data given and notation      

\bar X=25 represent the sample mean      

s=2 represent the standard deviation for the sample      

n=16 sample size      

\mu_o =24 represent the value that we want to test    

\alpha represent the significance level for the hypothesis test.    

t would represent the statistic (variable of interest)      

p_v represent the p value for the test (variable of interest)  

Confidence =0.95 or 95%

\alpha=0.05

State the null and alternative hypotheses.      

We need to conduct a hypothesis in order to determine if the mean is higher than 24, the system of hypothesis would be:      

Null hypothesis:\mu \leq 24      

Alternative hypothesis:\mu > 24      

We don't know the population deviation, so for this case is better apply a t test to compare the actual mean to the reference value, and the statistic is given by:      

t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}} (1)      

t-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".  

Calculate the statistic      

We can replace in formula (1) the info given like this:      

t=\frac{25-24}{\frac{2}{\sqrt{16}}}=2      

Calculate the P-value      

First we need to calculate the degrees of freedom given by:  

df=n-1=16-1=15  

Since is a one-side upper test the p value would be:      

p_v =P(t_{15}>2)=0.0320  

Conclusion      

If we compare the p value and the significance level given for example \alpha=0.05 we see that p_v so we can conclude that we reject the null hypothesis, and the true mean is significant higher than 24 years.      

3 0
3 years ago
A regular polygon had n-folds rotation symmetry. how many line of symmetry must it have
VashaNatasha [74]
6.6 Symmetries of Regular Polygons A Solidify Understanding Task A line that reflects a figure onto itself is called a line of symmetry. A figure that can be carried onto itself by a rotation is said to have rotational symmetry. A diagonal of a polygon is any line segment that connects non-consecutive vertices of the polygon. For each of the following regular polygons, describe the rotations and reflections that carry it onto itself: (be as specific as possible in your descriptions, such as specifying the angle of rotation) 1. An equilateral triangle 2. A square 3. A regular pentagon 4. A regular hexagon  
3 0
3 years ago
Read 2 more answers
What is the gcf for 18m 45mn?
Mice21 [21]
The answer is 9

9 times 2 = 18

9 times 5 = 45
8 0
3 years ago
Read 2 more answers
If a recipe calls for 0.800 kg of flour , about how many ounces of flour does it need? (1lb=16 oz, 1 lb = 0.454 kg )
stiks02 [169]
For this case we must take into account the following conversions of units:
 1lb = 16 oz
 1 lb = 0.454 kg
 Applying the conversion of units we have:
 (0.800) * (1 / 0.454) * (16/1) = 28.1938326
 Rounding off we have:
 28.2 oz
 Answer:
 
it needs 28.2 oz of flour
8 0
3 years ago
Read 2 more answers
The submarine was below sea level. b &lt; 0 A number line going from negative 2 to positive 2. Find the solutions of the inequal
qwelly [4]

Answer:

1. less than 0

2. negative

3. an infinite number of

Step-by-step explanation:

8 0
3 years ago
Read 2 more answers
Other questions:
  • Vinnie decorated 72 cookies in 36 minutes. How many cookies did he decorate per minute
    7·1 answer
  • 16,000 lb= __ T<br> 3 lb= __oz<br> lb= pounds T=tons OZ=ounce
    9·2 answers
  • What is the z score of a value that is 2.08 standard deviations greater than the mean
    12·1 answer
  • A fruit company delivers its fruit in two types of boxes: large and small. a delivery of 3 large boxes and 5 small boxes has a t
    14·1 answer
  • Can anybody help me answer this ASAP thank you
    7·1 answer
  • For 100​ births, P(exactly 55 ​girls)equals0.0485 and ​P(55 or more ​girls)equals0.184. Is 55 girls in 100 births a significantl
    15·1 answer
  • Find the slope of the line graphed in the diagram and choose the correct answer from the choices below. The two intercepts are (
    8·1 answer
  • The volume of a cylinder with height h and radius r is given by the formula below.
    10·1 answer
  • 6. There are 14 teams in a basketball league.
    7·2 answers
  • 15 = 15<br> True <br> 15 = 18 - 3<br><br> 15 + 3 = 11 + 7<br><br> 15 + 3 = 15 + 13
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!