Answers
Answer:
π
Step-by-step explanation:
Solving without L'Hopital's rule:
lim(x→0) sin(π cos²x) / x²
Use Pythagorean identity:
lim(x→0) sin(π (1 − sin²x)) / x²
lim(x→0) sin(π − π sin²x) / x²
Use angle difference formula:
lim(x→0) [ sin(π) cos(-π sin²x) − cos(π) sin(-π sin²x) ] / x²
lim(x→0) -sin(-π sin²x) / x²
Use angle reflection formula:
lim(x→0) sin(π sin²x) / x²
Now we multiply by π sin²x / π sin²x.
lim(x→0) [ sin(π sin²x) / x² ] (π sin²x / π sin²x)
lim(x→0) [ sin(π sin²x) / π sin²x] (π sin²x / x²)
lim(x→0) [ sin(π sin²x) / π sin²x] lim(x→0) (π sin²x / x²)
π lim(x→0) [ sin(π sin²x) / π sin²x] [lim(x→0) (sin x / x)]²
Use identity lim(u→0) (sin u / u) = 1.
π (1) (1)²
π
Solving with L'Hopital's rule:
If we plug in x = 0, the limit evaluates to 0/0. So using L'Hopital's rule:
lim(x→0) [ cos(π cos²x) (-2π cos x sin x) ] / 2x
lim(x→0) [ -π cos(π cos²x) sin(2x) ] / 2x
-π/2 lim(x→0) [ cos(π cos²x) sin(2x) ] / x
Again, the limit evaluates to 0/0. So using L'Hopital's rule one more time:
-π/2 lim(x→0) [ cos(π cos²x) (2 cos(2x)) + (-sin(π cos²x) (-2π cos x sin x)) sin(2x) ] / 1
-π/2 lim(x→0) [ 2 cos(π cos²x) cos(2x) + π sin(π cos²x) sin²(2x) ]
-π/2 (-2)
π
Related Questions
PLEASE ANSWER!!! I WILL MARK YOU AS BRAINLIEST!!!
Answers
Answer:
SAMPLE space=4
all outcomes:a,b,c,d
probability of each one is 1/4 or 0.25
2. probability for each one is 1 in 8
Step-by-step explanation:
Question 16 (Essay Worth 7 points)
Verify the identity.
tan (x + π/2) = -cot x
Answers
Step-by-step explanation:
We know that tan=sin/cos, so tan(x+π/2)=
[tex]\frac{sin(x+pi/2)}{cos(x+pi/2)}[/tex]
Then, we know that sin(u+v)=sin(u)cos(v)+cos(u)sin(v),
so our equation is then
[tex]\frac{sin(x)cos(\pi/2)+cos(x)sin(\pi/2)}{cos(x+\pi/2)} = \frac{cos(x)}{cos(x+\pi/2) }[/tex]
Then, cos(u+v)=cos(u)cos(v)-sin(u)sin(v), so our expression is then
[tex]\frac{cos(x)}{cos(x)cos(\pi/2)-sin(x)sin(\pi/2)} = \frac{cos(x)}{-sin(x)} = -cot(x)[/tex]
Of the following approximate conversions of Fahrenheit into Celsius, which is most accurate?
a. 14°F Almost-equals -10.8°C
b. 78°F Almost-equals 25.3°C
c. 106°F Almost-equals 41.7°C
d. 7°F Almost-equals -12.9°C
NVM it's B
Answers
Answer:
its B
Step-by-step explanation:
Answer:
b
Step-by-step explanation:
1- An ordered pair has coordinates that have the same sign. In which quadrant(s) could the point lie? EXPLAIN!!!
2 - Another ordered pair has coordinates that are opposites. In which quadrant(s) could the point lie? EXPLAIN!!!!
Answers
Answer:
as the sign of coordinates of different ordered pairs is as shown in the diagram so
ans. to 1 is quadrants one and three
ans. to 2 is quadrants two and four
how many different outfits can be made from 4 shirts, 4 pairs of pants, 2 pairs of shoes, and 3 hats
Answers
Answer:
Step-by-step explanation:
21
Which situation gives you the fewest number of possible outcomes?
A.
You choose 2 beads from a jar of 9 beads.
B.
You choose 4 cards out of 8 cards.
C.
You choose 5 people out of 6 people.
D.
You choose 4 socks out of 32 socks in a drawer.
Answers
Answer:
c you choose 5 out of 6 people
plus answer now Find 3/5 of 3000.
Answers
Answer:
1800
Step-by-step explanation:
HELP ME FIGURE THIS OUT
Answers
Answer:
your answer is -9,-4,0,2,7 so your answer is d
Step-by-step explanation:
by looking on the number line you can see that -9 is farthest to the left ,so meaning it would go first and 7 which is the farthest right meaning it should be last giving you the order -9 as first -4 as second and 0 as third and so on.
Justin did push-ups every weekday this week. He did 8 push-ups on Monday, 14 pushups on Tuesday 18 push
ups on Wednesday, 6 push-ups on Thursday, and 8 push-ups on Friday,
Find the median number of push-ups.
push-ups
Answers
Answer:
8
Step-by-step explanation:
The data set up lowest to highest is: 6, 8, 8, 14, 18. Median is the middle number of the data, so the answer would be 8
How much more ml will 12 cans hold when one can us 150ml
Answers
What is the interquartile range of the data represented by the box plot shown below?
Please! Help needed!!
Answers
Answer:
The interquartile range is 22.Step-by-step explanation:
The box plot shows all three quartiles, where the first one is 45, the second quartile is 50 and the third quartile is 67 (remember that there are only three quartiles in a dataset).
Now, the interquartile range is defined as the difference between the third and first quartile, which in this case is 67 - 45 = 22.
Therefore, the interquartile range is 22.
Tom bought a pair of shoes on sale for $60.29 the original price of $89.99 census what was the approximate percent of decrease that Tom bought the shoes for
please answer it's for hw
Answers
explanation: 67% ig :)
0.85 converts to a fraction with a numerator of 85 and a denominator of
Answers
Answer:
denominator is 100
Answer:
The demominator is 100
Step-by-step explanation:
0.85 is pronounced eighty-five hundreths therefore the denominator is 100!
I NEED ANSWER ASAP!
Chiara has 26 coins that equal 34 cents. All the coins are pennies, p, and nickels, n. How many nickels and pennies does Chiara have? Use the table to guess and check.
1. 20 pennies and 6 nickels
2. 21 pennies and 5 nickels
3. 23 pennies and 3 nickels
4. 24 pennies and 2 nickels
Answers
Answer:
#4
Step-by-step explanation:
p+n=26
.01p+.05n=.34
graph it out by desmos...?
you'll get the coordinates (24,2)
The number of pennies and nickels chiara has is 24 pennies and 2 nickels; the correct option is 4
What is the solution for an equation?The solution of an equation refers usually to the values of the variables involved in that equation which if substituted in place of those variable would give a true mathematical statement.
Let;
p = the number of penny coins
n = the number of nickel coins
Remember;
1 penny = $0.01
1 nickel = $0.05
Chiara has 26 coins that equal 34 cents
so, p + n = 26
isolate the variable x;
p = 26 - n...equation(1)
34 cents=$0.34
0.01p + 0.05n = 0.34
Multiply by 100 both sides;
1p + 5n =$34
isolate the variable x;
p = 34 -5n...equation(2)
Create a table to guess;
Assume different values of n and determine the value of p in equation 1 and equation 2
The solution is when the value of p in the equation 1 must be equal to the value of p in equation 2
For n=1;
equation 1; p = 26-1 = 25
equation 2; p = 34-5(1)= 29
29 is not equal to 25
For n=2
equation 1; p= 26-2 = 24
equation 2; p= 34-5(2) = 24
24=24
Hence, the answer is (2,24)
Learn more about solving equations here:
https://brainly.com/question/13072448
#SPJ6
Sally completely solves at least 7 questions per hour and at most 10 questions per hours. Based on this information, what is a possible number of questions that Sally can completely solve in 5 hours?
Answers
Answer:
Minimum-35
Maximum-50
Step-by-step explanation:
Sally can solve 35 questions in 5 hours and at most 70 question in 5 hours.
What is mean by Ratio?
A ratio indicates how many times one number contain in another number. The ratio of two number is written as x : y, which is equivalent to x/y.
Where, x and y are individual amount of two quantities.
And, Total quantity gives after combine as x + y.
Given that,
Minimum number of questions sally can solve in 1 hour = 7
Maximum number of questions sally can solve in 1 hour = 10
Now minimum of of question sally can solve in 5 hour,
= 5x7
= 35 questions
And Maximum number of question she can solve in 5 hour,
= 5x10
= 50 questions.
Hence,
In 5 hours she can solve,
Minimum 35 questions
Maximum 50 questions.
Learn more about the ratio visit:
https://brainly.com/question/12024093
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Solve these equations. If possible, reduce your answers.
3/4x=−1 1/4
Answers
Answer:
-1 2/6
Step-by-step explanation:
Answer:
Step-by-step explanation:
What is the range of the function y=e* + 1 graphed below?
Answers
[tex]$f: \mathbb{R} \rightarrow \mathbb{R}, f(x) = e^{x} + 1$[/tex]
[tex]\text{Range}(f) = \{y \in \mathbb{R} \vert x \in \mathbb{R}, f(x) = y\}[/tex]
[tex]\text{Suppose $f(x) = a$}\\e^{x} + 1 = a \implies e^{x} = a - 1 \implies x = \ln(a-1)\\[/tex]
[tex]\text{For the logarithm to be real, it must be true that $a - 1 > 0$}\\\text{But then we have that } a > 1. \implies \boxed{\text{Range}(f) = (1, \infty)}[/tex]
Answer:
The answer is D. y > 1 on edge
Andre is preparing for the school play and needs to paint the cardboard castle backdrop that measures feet by 6 feet.
How much cardboard does he need to paint?
If one bottle of paint covers an a area of 40 square feet, how many bottles of paint does Andre need for his backdrop
Answers
Answer:
a. 85.5 sq ft
b. 3 bottles
Step-by-step explanation:
a. The cardboard is rectangular in shape. How much he needs to paint is the same as the cardboard that he needs to paint.
The area of a rectangle is given as:
A = L * B
where L = length and B = breadth
Therefore, the area of the cardboard having dimensions 14 1/4 feet by 6 feet is:
A = 14 1/4 * 6
A = 57 / 4 * 6 = 85.5 sq feet
He needs to paint 85.5 sq feet of the cardboard.
b. On bottle of paint covers 40 sq feet. The number of bottles he will need for his backdrop will be the area he needs to paint divided by the amount of paint that can cover 40 sq feet:
85.5 / 40 = 2.14
Since the number of bottles he will need is a whole number and its greater than 2, then it must be 3.
To paint the entire cardboard, he must have 3 bottles of paint.
Answer:
75 feet
Step-by-step explanation:
work out 4.50 ÷ 0.05
Answers
Answer:
90
Step-by-step explanation:
If it takes two skilled workers 2 weeks to
remodel two bathrooms, how long will it take
10 skilled workers to remodel 10 bathrooms?
Answers
The diameter of a circle is 4 in. Find its area in terms of π\pi π.
Answers
The formula for area of a circle is r² x π.
Since we know the diameter, we can firstly divide by 2 to find our radius.
4/2 = 2
Now that we know the radius, we can substitute it in and square the value.
(2)² x π
(4) x π
Since we're finding the area in terms of pi, we won't multiply by pi, giving us the answer of 4π.
Therefore, the area of the circle is equal to 4π.
Hope this helped! :)
Answer:
If the diameter of the circle is 4 in. then the radius of the circle is 2
the formula for the circle area is
A = πr^2
r = radius
A = π * 2 * 2
A = 4πA = 12.57 if solved
Step-by-step explanation:
Given that the y intercept is 3 and the slope is -2/3 what is the equation of the line?
a) y=2/3x+3 b) y= -2/3+3
c) y=3x-2/3 d) y=3x+2/3
Answers
Answer:
B y=-2/3x+3
Step-by-step explanation:
y=mx+b
y=-2/3x+3
Answer:
Step-by-step explanation: y= -2/3x+3
EASY 30 POINTS! The population of the town of White Plains has decreased by 2.2 percent per year since 1970. In 1995, the town had a population of approximately 1,200 people. Which of the following functions represents the town’s population, P, y years since 1995?
P(y) = 1,200(−2.2)y
P(y) = 1,200(0.978)y
P(y) = 1,200(−2.2)y − 25
P(y) = 1,200(0.978)y − 25
GIVING BRANLIEST
Answers
Answer:
B
Step-by-step explanation:
Two groups of subjects were divided in an experimental study. One group was given a drug to help speed up
their metabolism and result in weight loss. The other group was given a placebo (a pill that looks identical to the one given to the other group, but without the weight loss drug). After a month of the experiment, the weight loss of each individual in each of the two groups were measured. In general, people in the group who took the metabolism drug did lose more weight, although there were differences in the amount each lost. There are two main types of variability occurring in this study. Describe each type below in the context of this study.
Induced Variability-
Natural Variability-
Answers
Answer:
Step-by-step explanation:
Two groups of subjects were divided in an experimental study. One group was given a drug to help speed up
their metabolism and result in weight loss. The other group was given a placebo (a pill that looks identical to the one given to the other group, but without the weight loss drug). After a month of the experiment, the weight loss of each individual in each of the two groups were measured. In general, people in the group who took the metabolism drug did lose more weight, although there were differences in the amount each lost. There are two main types of variability occurring in this study. Describe each type below in the context of this study.
Induced Variability-
Natural Variability-
------------------------------------------------------------------------
Induced Variability - It is the kind of variability that is introduced by the one conducting the experiment. The decision to give one group the drug and othe other group a placebo was deliberate to check the merit of the drug on its own and was introduced by the one conducting the study. Hence the, the observation that the group that took the drug lost more weight as compared to the group that took the placebo is an induced variability.
Natural Variability - It is classified as a natural fluctuation in the observations that is not influenced by the one conducting the study. Even though the group that took the drug lost more weight, the exact amount that each of them lost was not constant. Hence this was a variability that did not introduce itself through the one conducting the study, and rather was unexplained and natural. Hence it is a natural variability.
-12 + 18. = 70
complete the square
Answers
What is the solution to the inequality?
-2x - 4 ≤ 11
Thank you!
Answers
Answer:
x ≥ -15/2
Step-by-step explanation:
-2x - 4 ≤ 11
Add 4 to each side
-2x - 4+4 ≤ 11+4
-2x ≤ 15
Divide each side by -2, remembering to flip the inequality since we are dividing by a negative
-2x/-2 ≥ 15/-2
x ≥ -15/2
Answer:
x≤-7.5
Step-by-step explanation:
-2x - 4 ≤ 11
Add 4 to both sides
-2x≤15/-2
Divide
x≤-7.5
Annie has $18.00 to spend at the grocery store. She is going to buy bags of nuts that cost $5.70 each and one box of cookies that costs $4.25.Write and solve an inequality that models this situation and could be used to determine the maximum number of bags of nuts, n, Annie can buy. (Show all your work)
Answers
Answer: She can buy up to 2 bags of nuts
Step-by-step explanation:
Hi, to answer this question we have to write an inequality:
The product of the number of bags of nuts bought (n) and the price per bag (5.70); plus the price of one box of cookies (4.25) must be less or equal to Annie's money (18).
5.70n +4.25 ≤18
Solving for x:
5.70n ≤18 -4.25
5.70n ≤13.75
n ≤13.75/5.70
n ≤2.41
n ≤2 (rounded)
She can buy up to 2 bags of nuts
Answer:2
Step-by-step explanation:
Name a quadrilateral with exactly two pairs of congruent angles.
Answers
parallogram it has exactly 2 pairs of congruent angles
Answer:
Parallelogram.
Step-by-step explanation:
Find the average of 23 and −7.
Answers
Answer:
8
Step-by-step explanation:
23 + (-7) =16
16/2 = 8
Answer: 8
Step-by-step explanation:
23+(-7) = 16/2 = 8
Two fire-lookout stations are 100 miles apart, with station A directly south of station B. Both stations spot a fire. The bearing of the fire from station A is Upper N 55 degrees Upper EN55°E and the bearing of the fire from station B is Upper S 60 degrees E.S60°E. How far, to the nearest tenth of a mile, is the fire from each lookout station?
Answers
Answer:
The distance from station A to the fire is 90.4 miles
The distance from station B to the fire is 95.6 miles
Step-by-step explanation:
Given that 2 fire stations are 100 miles apart.
Station A = N 55°E
Station B = S 60°E
To find the distance of the fire from each station, let's use the law of sines,
From the diagram, our values are:
A = 55°
B = 60°
Let's take C as point of fire.
Since the total sum of angle in a triangle is 180,
A+B+C=180
C = 180 - A - B
C = 180 - 55 - 60 = 65°
For station A.
[tex] \frac{Sin A}{a} = \frac{Sin C}{c} [/tex]
[tex]\frac{Sin55}{a} = \frac{Sin65}{100}[/tex]
[tex]a = \frac{Sin55 * 100}{Sin65}[/tex]
[tex]a = 90.38[/tex]
The distance from station A to the fire is 90.4 miles
For station B:
[tex] \frac{Sin B}{b} = \frac{Sin C}{c} [/tex]
[tex]\frac{Sin60}{b} = \frac{Sin65}{100}[/tex]
Solving for b, we have:
[tex]a = \frac{Sin60 * 100}{Sin65}[/tex]
[tex]a = 95.6[/tex]
The distance from station B to the fire is 95.6 miles