Answer:
[tex]\displaystyle \frac{d}{dx}[\tan x] = \sec^2 x[/tex]
General Formulas and Concepts:
Pre-Calculus
Trigonometric IdentitiesCalculus
Differentiation
DerivativesDerivative NotationDerivative Rule [Quotient Rule]: [tex]\displaystyle \frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}[/tex]
Step-by-step explanation:
*Note:
This is a known trigonometric derivative.
Step 1: Define
Identify
[tex]\displaystyle y = \tan x[/tex]
Step 2: Differentiate
Rewrite [Trigonometric Identities]: [tex]\displaystyle y = \frac{\sin x}{\cos x}[/tex]Derivative Rule [Quotient Rule]: [tex]\displaystyle y' = \frac{(\sin x)'\cos x - \sin x(\cos x)'}{\cos^2 x}[/tex]Trigonometric Differentiation: [tex]\displaystyle y' = \frac{\cos^2 x + \sin^2 x}{\cos^2 x}[/tex]Rewrite [Trigonometric Identities]: [tex]\displaystyle y' = \frac{1}{\cos^2 x}[/tex]Rewrite [Trigonometric Identities]: [tex]\displaystyle y' = \sec^2 x[/tex]Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
The derivative of tanx in calculus is sec^2x, which represents the slope of the curve at any point x for the function tanx.
Explanation:The derivative of a function measures how the function changes as its input changes. In calculus, the derivative is a concept that describes the rate at which a quantity changes.
In the context of your question, the derivative of tanx or tangent x is sec^2x or secant squared x. This is a standard result that is usually memorized in calculus courses and can be derived using the limit definition of the derivative and trigonometric identities.
To put it in simpler terms, if tanx were the equation of a curve, the derivative (sec^2x) would represent the slope of that curve at any point x. This is key to understanding how the function behaves at different points and predicting its future behavior.
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1. if x and y are both even whole numbers. what is a possible solution for x + y?
a. 5
b. 9
c. 4.6
d. 8
2. assume n and m are both even whole numbers and m > n. what is a possible solution for m - n?
a. 16
b. 1/2
c. 0
d. -2
1. The possible solution for x + y is 8 ⇒ d
2. The possible solution for m - n is 16 ⇒ a
Step-by-step explanation:
Lets explain some notes about the even numbers
Their sum is evenTheir difference is evenTheir product is even1.
∵ x and y are both even whole numbers
∴ Their sum is even whole number
∵ 5 and 9 are odd numbers ⇒ impossible
∵ 4.6 is not a whole number ⇒ impossible
∵ 8 is an even whole number ⇒ possible
∴ x + y could be 8
The possible solution for x + y is 8
2.
∵ n and m are both even whole numbers
∵ m > n
∴ m - n must be positive even whole number
∵ 1/2 is not a whole number ⇒ impossible
∵ 0 is not a positive number ⇒ impossible
(zero is not positive nor negative)
∵ -2 is not a positive number ⇒ impossible
∵ 16 is a positive even whole number ⇒ possible
∴ m - n could be 16
The possible solution for m - n is 16
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According to Okun's law, if the unemployment rate goes from 6% to 2%, what
will be the effect on the GDP?
A. It will decrease by 11%.
B. It will increase by 5%.
c. It will decrease by 5%.
D. It will increase by 11%.
Answer: D
Step-by-step explanation:
Okun's law says that a country's gross domestic product (GDP) must grow at about a 4% rate for one year to achieve a 1% reduction in the rate of unemployment.
Option D is correct.
Since, According to Okun's law a country's gross domestic product (GDP) must grow at about a 4% rate for one year to achieve a 1% reduction in the rate of unemployment.
So, if the unemployment rate goes from 6% to 2%, that is reduce by 4%
Therefore, GDP is increase by 4(4 %) = 16 %
Thus, most appropriate answer is option D, that is GDP will increase by 11% .
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Describe the transformations.
Given the function [tex]f(x)[/tex]
[tex]f(x)+k[/tex] — will shift the graph vertically by k
[tex]f(x-k)[/tex] — will shift the graph horizontally by k
[tex]kf(x)[/tex] — will stretch the graph vertically by k
[tex]f(kx)[/tex] — will stretch the graph horizontally by k
Katy bought a candy bar and shared it with her two sisters. She gave her youngest sister 1/4 of the candy bar. Her older sister received 1/3 of the candy bar. How much of the candy bar did Katy keep?
Answer:
Step-by-step explanation:
Since the candy bar is just one
Giving 1/4 of it to her younger sister make her left with 3/4 of the candy bar i.e 1 - 1/4 = 3/4
Then giving 1/3 of the left over to her elder sister it becomes 3/4 * 1/3 =1/4
Then the remaining candy bar will be 3/4 - 1/4 = 1/2
Then she will be left with 1/2 i.e half of the candy bar.
-3х + 40 = 10
I don’t know how to solve
Answer:
Step-by-step explanation:
I NEED HELP ASAP!!!!
the average number 1 0 1 5 6 3 4
Answer: 2
Step-by-step explanation: Edge 2023
Final answer:
To find the average number from a list of numbers, add up all the numbers and then divide the sum by the total number of numbers.
Explanation:
The question asks for the average number from a given list of numbers. To find the average, you add up all the numbers in the list and then divide the sum by the total number of numbers in the list.
In this case, the average of the numbers 1, 0, 1, 5, 6, 3, and 4 can be found by adding those numbers together (1 + 0 + 1 + 5 + 6 + 3 + 4) and then dividing the sum by 7 (the total number of numbers in the list). So, the average is 4.
A coffee shop served hot and cold coffee to a total of 340 people. 45% of the people chose cold coffee.
How many people chose hot coffee?
Answer:
187 people
Step-by-step explanation:
340 people multiplied by 0.45 or 45% is 153. Subtract 153 from 340 and you get 187 people.
Answer:
187 people
Step-by-step explanation:
340 people multiplied by 0.45 or 45% is 153. Subtract 153 from 340 and you get 187 people.
What is a real world situation for the problem 42.48 Divided by 25
Answer:
At the donut shop, Karley and 24 of her co-workers have split the tip jar. There is $42.48 in the tip jar. How much money does each worker get from the jar?
Help me please, I’ll give brainliest
Answer:
x=11
Step-by-step explanation:
since ad bisects bac, bad=dac
32=3x-1
+1 +1
3x=33
/3 /3
x=11
WILL MAKE BRAINLEST!! Which equation best represents the line??
A. Y=1/3X-1
B. Y=3X-1
C. Y= -X+ 1/3
D. Y= 3X+1
Answer:
B is the answer. ........
Answer:
B
Step-by-step explanation:
it intercepts the y line at -1 and the rise over run is 3
How Do You Write The Expression For Triple 9 Then Multiply 8
In order to find the height of a cliff, you stand at the bottom of the cliff, walk 60 feet
from the base, and place a mirror on the ground. Then you face the cliff and step back
5 feet so that can see the top of the cliff in the mirror. Assuming your eyes are 6 feet
above the ground, explain how to use this information to find the height of the cliff.
(The angles marked congruent are congruent because of the nature of the reflection of
light in a mirror.)
Q5ft
60 ft
K
Mirror
Answer: 72 ft
Step-by-step explanation:
We can solve this problem using geometry, if we draw two righ triangles (figure attached):
-One formed by you, assuming you are 6 ft tall, and the ground
-The other formed by the height of the cliff, the bottom of the cliff and the ground.
Both triangles are congruent, having the mirror as a common vertex, hence we can use the following relation to find :
[tex]\frac{6 ft}{y}=\frac{5 ft}{60 ft}[/tex]
Isolating :
[tex]y=\frac{(60 ft)(6 ft)}{5 ft}=72 ft[/tex] This is the height of the cliff
The calculated height of the cliff is 72 feet.
To determine the height of the cliff, we use a mirror and the property of the reflection of light. Here are the steps to solve the problem:
Place the mirror 60 feet from the base of the cliff and stand back 5 feet from the mirror.Since your eyes are 6 feet above the ground, the light reflecting from the mirror at the point where you see the top of the cliff will create similar triangles based on the reflection.The triangles formed involve your height of 6 feet over the 5 feet back from the mirror and the unknown height of the cliff over the 60 feet from the mirror.Set up the proportion from the similar triangles:height of cliff / 60 feet = 6 feet / 5 feet.
5. Solve for the height of the cliff:
height of cliff = (6 feet / 5 feet) * 60 feet = 72 feet.
Thus, the height of the cliff is 72 feet.
Ryan’s heart beats 9 times per 10 seconds while Ryan is resting. About how many times would Ryan’s heat beat during 3 minutes of rest.
What is the expression for triple 9 then multiply 8
Answer:
3R
Step-by-step explanation:
Find the circumcenter of the triangle ABC.
A( - 4,3), B(5,7), C(-4,7).
Answer:
Circumcenter O (0.5 , 5)
Step-by-step explanation:
Circumcenter O (x , y)
OA = OB = OC
OA² = (x + 4)² + (y - 3)²
OB² = (x - 5)² + (y - 7)²
OC² = ( x + 4)² + (y - 7)²
OA² = OC² : (x + 4)² + (y - 3)² = ( x + 4)² + (y - 7)²
(y - 3)² = (y - 7)² y² - 6y + 9 = y² -14y + 49
8y = 40
y = 5
OB² = OC² : (x - 5)² + (y - 7)² = ( x + 4)² + (y - 7)²
x² - 10x + 25 = x² + 8x + 16
18x = 9
x = 0.5
Circumcenter O (0.5 , 5)
The distance between Matt's house and his school is 3 inches on the map. The actual distance is 1.5 miles. What is the
of the map?
1 inch = 2 miles
1 inch = 0.5 mile
3 inches = 1 mile
1.5 inches = 1 mile
Answer:
b
Step-by-step explanation:
What is gcf of 24, 60, and 36
Answer:
12
Step-by-step explanation:
Darcy’s house is 6 times as far from school as her friend’s house. If her friend’s house is 3/4 mile from school, how many miles from school is Darcy’s house?
Please help ASAP will give BRAINLIEST ANSWER!!!!!!!
Answer:A $35 decrease followed by a $25 increase is less than the original.
A 50% decrease then 33.333% increase is the same.
A 30% increase then 15% decrease is more.
A 50% decrease then 100% increase is the same.
A 70% increase then 50%decrease is less.
Step-by-step explanation:
f(x)=−x+2, find f(-5)f(−5).
To calculate f(-5) for the function f(x)=-x+2, we substitute -5 in place of x, which gives us f(-5) = 7.
Explanation:To find the value of f(-5) for the function f(x)=−x+2, we simply substitute x with -5:
f(-5) = −(-5) + 2
Now, we perform the operations:
f(-5) = 5 + 2
f(-5) = 7
Therefore, the answer is 7.
Alexis and 3 friends are at the mall.each person buys a pretzel for 4$, sauce for $1, and a drink for $2. write and expression for the total and find the total cost.
Answer:
(4+1+2)3= 21
Step-by-step explanation:
you add the cost of snacks then multiply by 3 because there are 3 people to get total cost.
4+1+2=7 7*3=21Answer:
4+4+4+1+1+1+2+2+2=25
Colin invests £3500 into his bank account.
He receives 5% per year compound interest.
How much will Colin have after 6 years?
Give your answer to the nearest penny where appropriate.
Answer: 4,690.33
Step-by-step explanation:
The formula for calculating amount is given by :
A = P [tex](1+r)^{n}[/tex]
Where :
A = amount = ?
P = Principal = 3500
r = rate = 0.05
n = number of years = 6
substituting the given values into the formula , we have
A = 3500[tex](1+0.05)^{6}[/tex]
A = 3500 [tex](1.05)^{6}[/tex]
A = 3500 (1.340095640625)
A = 4690.3347421875
A = 4,690.33
Final answer:
To calculate Colin's investment with compound interest after 6 years, we use the formula for compound interest and determine that he will have £4690.33 to the nearest penny.
Explanation:
To calculate how much Colin will have after 6 years with compound interest, we use the formula for compound interest:
A = P(1 + r/n)nt
Where:
A is the amount of money accumulated after n years, including interest.
P is the principal amount (the initial amount of money).
r is the annual interest rate (decimal).
n is the number of times that interest is compounded per year.
t is the time the money is invested for, in years.
In Colin's case, we have:
P = £3500
r = 5% or 0.05
n is not mentioned, so we assume it is compounded once per year, thus n = 1
t = 6 years
Plugging these values into the formula:
A = £3500(1 + 0.05/1)1×6 = £3500(1 + 0.05)6
A = £3500 × (1.05)6 = £3500 × 1.340095
A = £4690.3325
To the nearest penny, Colin will have
£4690.33 after 6 years.
Your neighbor has decided to enlarge his garden. The garden is rectangular with width 6 feet and length 15 feet. The new garden will be similar to the original one, but will have a length of 35 feet. Find the perimeter of the original garden and the enlarged garden.
The perimeter of original garden is 42 feet
The perimeter of enlarged garden is 82 feet
Solution:
The perimeter of rectangle is given as:
Perimeter = 2(length + width)
Original Garden:Given that,
The garden is rectangular with width 6 feet and length 15 feet
Length = 15 feet
Width = 6 feet
Therefore, perimeter is given as:
Perimeter = 2(15 + 6)
Perimeter = 2(21)
Perimeter = 42
Thus perimeter of original garden is 42 feet
Enlarged Garden:The new garden will be similar to the original one, but will have a length of 35 feet.
Length = 35 feet
Width = 6 feet
Therefore, perimeter is given as:
Perimeter = 2(35 + 6)
Perimeter = 2(41) = 82
Thus perimeter of enlarged garden is 82 feet
2. Tickets to a play cost $8 each. Write an
expression that gives the ticket cost in dollars
for a group of g girls and b boys.
Expression that gives the ticket cost in dollars for a group of g girls and b boys is (8g + 8b) dollars
Solution:
Given that tickets to play cost $ 8 each
To find: Expression that gives the ticket cost in dollars for a group of g girls and b boys
From given,
cost of 1 ticket = $ 8
Number of girls = g
Number of boys = b
Thus the expression can be framed for total cost as:
Total cost = (number of girls + number of boys)(cost of 1 ticket)
[tex]Total\ Cost = (g + b) \times 8\\\\Total\ Cost = 8g + 8b[/tex]
Thus the expression that gives total cost is (8g + 8b) dollars
NEED HELP NOW
Shawn got a haircut at the barber shop. It cost $11, and he left a $3 tip.
What percent tip did he leave the barber?
Answer: He left a 27.27272727% tip
Simplify: (5x^6y^-4)^3(2x^8y^3)
Answer:
250x^26/y^9
Step-by-step explanation:
Trevin puts 8 gallons of gas in his car and pays $22, ate this rate what is the cost for 9 gallon of gas
Answer:
$24.75
Step-by-step explanation:
Find the unit rate for 1 gallon, to dothis divide $22 by the 8 gallons of gas. 22/8 is $2.75. now that we know the unit rate, multiply it by 9 for our answer. 2.75*9=24.75,the cost for 9 gallons.
Trevin puts 8 gallons of gas in his car and pays $22. The cost of 9 gallons of gas at this rate is $24.75
Word problems in mathematics involve a careful understanding of the question. It is usually written or expressed in one or more sentences and it requires pupils to apply their arithmetic skills to a real-life issue.
From the information given:
Trevin put 8 gallons of gas in his car and pays $22 for iti.e
8 gallons = $22We are to determine the rate he will pay for 9 gallons. i.e.
9 gallons = xFrom the above two expressions, by cross multiplying, we have:
[tex]\mathbf{x = \dfrac{9 \ gallons \times \$22 }{8 \ gallons}}[/tex]
x = $24.75
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a dragonfly can beat its wings 30 times per second. write an equation in slope-intercept form that shows the relationship between flying time in seconds and the number of times that dragonfly beats its wings
Helppppppp let’s get to 100!!!!
Answer: 3
Step-by-step explanation:
10 had sprinkles and 2 did not which mean there are 12.
So would it be 3 that not have a sprinkles?
I think so.