Explain how the order of operations determines how you simplify a numeric expression. Be sure to explain the rules for addition, subtraction, multiplication, division, and the use of grouping symbols. Use examples to illustrate your points.
What is the derivative of cos(pi*t/4) ?
Answer:
[tex]\displaystyle \frac{dy}{dx} = \frac{-\pi}{4} \sin \bigg( \frac{\pi t}{4} \bigg)[/tex]
General Formulas and Concepts:
Calculus
Differentiation
DerivativesDerivative NotationDerivative Property [Multiplied Constant]: [tex]\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)[/tex]
Basic Power Rule:
f(x) = cxⁿf’(x) = c·nxⁿ⁻¹Derivative Rule [Chain Rule]: [tex]\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)[/tex]
Step-by-step explanation:
Step 1: Define
Identify
[tex]\displaystyle y = \cos \bigg( \frac{\pi t}{4} \bigg)[/tex]
Step 2: Differentiate
Trigonometric Differentiation [Derivative Rule - Chain Rule]: [tex]\displaystyle y' = -\sin \bigg( \frac{\pi t}{4} \bigg) \cdot \frac{d}{dt} \bigg[ \frac{\pi t}{4} \bigg][/tex]Rewrite [Derivative Property - Multiplied Constant]: [tex]\displaystyle y' = -\sin \bigg( \frac{\pi t}{4} \bigg) \cdot \frac{\pi}{4} \frac{d}{dt}[t][/tex]Basic Power Rule: [tex]\displaystyle y' = \frac{-\pi}{4}\sin \bigg( \frac{\pi t}{4} \bigg)[/tex]Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
If two lines intersect and form 4 right angles, then ___________.
A.the lines are parallel
B.the lines are perpendicular
C.the angles are complementary
D.the lines are not coplanar
Two lines that intersect and form four right angles are perpendicular to each other. This is because they cross at a single point creating four 90 degree angles. Therefore, the lines are not parallel, the angles are not complementary, and their coplanar status is irrelevant.
Explanation:If two lines intersect and form four right angles, it means that the lines are perpendicular to each other. In geometry, when two lines are perpendicular, they intersect at one point forming four right angles. A right angle is an angle of 90 degrees, which is the total division of a straight angle. Therefore, the two lines are not parallel because parallel lines do not intersect. The angles would not be complementary because complementary angles are two angles that add up to 90 degrees. Lastly, the lines being coplanar or not has no bearing on the angles they form when they intersect.
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Two lines intersecting to form four right angles are perpendicular, which is reflected in B. the lines are perpendicular. Parallel lines don't intersect, complementary angles are about their sum, and not coplanar doesn't relate to angles formed by intersection.
Explanation:When two lines intersect and form 4 right angles, they possess a special relationship known as perpendicularity. These perpendicular lines meet to form 90-degree angles, also referred to as right angles. This fact is grounded in basic geometric principles.
The correct answer to the question is B. the lines are perpendicular. Choosing any of the other options would not accurately describe the relationship between the lines. Parallel lines do not intersect, complementary angles sum up to 90 degrees but do not describe the relationship between lines, and being not coplanar implies the lines do not lie in the same plane which is irrelevant to the intersection forming right angles.
It is important to recognize that according to Theorem 1, two straight lines cannot intersect in more than one point and Theorem 2 states that vertical angles, which are the angles opposite each other when two lines intersect, are equal. All this supports the fact that if the lines intersect in such a way to form 4 right angles, they are indeed perpendicular.
Deandre has put his coins into 3 stacks. each stack has the same number of coins. there are 12 coins total. how many coins are in each stack?
(6 points)
2. Solve each given equation and show your work. Tell whether it has one solution, an infinite number of solutions, or no solutions, and identify each equation as an identity, a contradiction, or neither.
(a) 6x + 4x - 6 = 24 + 9x
(b) 25 - 4x = 15 - 3x + 10 - x
(c) 4x + 8 = 2x + 7 + 2x - 20
Answer:
Which is greater one mile or one kilometer?
what is the multiple zero and multiplicity of x^3+2x^2+x
Answer:
Step-by-step explanation:
It is 2x by the third power because i dont know lol
Which of the following is a solution to the equation c+(4-3c)-2=0
g(x) = arcsin(4x + 8)
Find the domain of this function.
How to solve: 52 is 65% of what number?
The number 52 is 65% of the number 80.
What is Percentage?Chance formula is used to find the quantum or share of commodity in terms of 100. In its simplest form, percent means per hundred. To express a number between zero and one, chance formula is used. It's defined as a number represented as a bit of 100. It's denoted by the symbol, and is majorly used to compare and find out rates.
We have to find 65% of what number is 52.
let the number be x.
So, 65% of x =52
65/100 x = 52
0.65 x = 52
x= 52/ 0.65
x= 80.
Thus, 65% of 80 is 52.
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The solution to -16 + x = -30 is -14.
True
False
Answer:
True
Step-by-step explanation:
[tex]x = - 30 + 16 \\ \\ or \: \: x = - 14[/tex]
Find the area of the largest isosceles triangle that can be inscribed in a circle of radius 6 inches.
Anthony's birthday, his mother is making cupcakes for his 12 friends at his daycare. the recipe calls for 3 cups of flour. this recipe makes 2 1/2 dozen cupcakes. anthony's mother has only 1 cup of flour. is there enough flour for each of his friends to get a cupcake? k
rewrite each expression more simply 5x+4x
What does the common difference or common ratio mean??
in mathematics, the common difference or we can call it the rate of change is the amount that we are constantly changing by either adding or subtracting to get to our next term. In other words, it’s consecutive terms increasing or decreasing.
Which function represents the graph y=3(1/2)^x ?
Function a
Function b
Function c
Function d
Answer:
Option B.
Step-by-step explanation:
Consider the below figure attached with this question.
The given function is
[tex]y=3(\frac{1}{2})^x[/tex] .... (1)
We need to find the graph of given function.
The general form of an exponential function is
[tex]y=ab^x[/tex] .... (2)
where, a is initial value and b is growth or decay factor.
If 0<b<1, then it is a decreasing function.
If b>1, then it is an increasing function.
On comparing (1) and (2) we get
[tex]a=3, b=\frac{1}{2}[/tex]
a=3, it means y-intercept of the function is 3.
[tex]b=\frac{1}{2}[/tex], it means it is a decreasing function.
The table of values is
x y
-2 12
-1 6
0 3
1 1.5
2 0.75
Plot these ordered pairs on a coordinate plane and connect then by a hand curve. Graph B represents the given function.
Therefore, the correct option is B.
Three roots of a fifth degree polynomial function f(x) are –2, 2, and 4 + i. Which statement describes the number and nature of all roots for this function?
f(x) has two real roots and one imaginary root.
f(x) has three real roots.
f(x) has five real roots.
f(x) has three real roots and two imaginary roots.
Answer: The correct option is f(x) has three real roots and two imaginary roots.
Explanation:
It is given that the roots of fifth degree polynomial function are -2, 2 and 4+i.
Since he degree of f(x) is 5, therefore there are 5 roots of the function either real or imaginary.
According to the complex conjugate root theorem, if a+ib is a root of a polynomial function f(x), then a-ib is also a root of the polynomial f(x).
Since 4+i is a root of f(x), so by complex conjugate rot theorem 4-i is also a root of f(x).
From the the given data the number of real roots is 2 and the number of 2. The number of complex roots is always an even number, so the last root must be a real number.
Therefore, the correct option is f(x) has three real roots and two imaginary roots.
A population of bacteria is introduced into a culture. the number of bacteria P can be modeled by P=500(1+4t/(50+t^2 )) where t is time in hours. Find the rate of change in population when t=2 ...?
Find the average of the function over the given interval and all values of x in the interval for which the function equals its average value. (Round your answer to three decimal places.)
f(x) = 8 cos(x),
(0, π/6)
...?
How many solutions are there to the equation below?
17x- 8 = 3x + 16
A. 1
B. 0
C. infinitely many
Answer:
only one solution
Step-by-step explanation:
The given equation is
17x- 8 = 3x + 16
Add 8 to both sides
17x = 3x + 24
Subtract 3x to both sides
14x = 24
Divide both sides by 14
x = 24/14
x = 12/7
Since, we got only one value of x.
Therefore, the equation has only one solution.
A is correct option.
fourteen boxes of candle were sold for 91.00.each box contains 5 candles. what is the cost of 1 candle
The cost of one candle can be found by dividing the total cost ($91.00) by the total number of candles sold (70), resulting in $1.30 per candle.
To find the cost of 1 candle, first calculate the total cost of all the candles sold. Fourteen boxes were sold for $91.00 in total. Since each box contains 5 candles, you can find the total number of candles by multiplying 14 boxes by 5 candles per box.
14 boxes × 5 candles per box = 70 candles total.
Now, to find the cost of one candle, divide the total cost by the total number of candles.
$91.00 / 70 candles = $1.30 per candle
Therefore, the cost of one candle is $1.30.
Write this function in vertex form, and Identify its vertex x^2-4x-17
Which is bigger 4.953 or 4.951?
4.953 is larger than 4.951 because when comparing the digits from left to right, the thousandths place is different, where 3 is greater than 1.
To determine which number is larger, we compare them digit by digit starting from the left:
Both numbers have the same digit in the units place: 4.
Both numbers have the same digit in the tenths place: 9.
Both numbers have the same digit in the hundredths place: 5.
In the thousandths place, 4.953 has a 3, and 4.951 has a 1.
Since 3 is greater than 1, 4.953 is larger than 4.951.
How do i find an angle that is complementary to theta?
...?
Through: (-2,4), parallel to y = -3/2x+3
To find a line parallel to y = -3/2x+3 through the point (-2,4), we use the point-slope form of a line using the given slope and coordinates.
Explanation:To find a line parallel to a given line, we need to note that parallel lines have the same slope. The given equation is in the form y = mx + b, where m is the slope. In this case, the slope of the given line is -3/2. So, our parallel line will also have a slope of -3/2. Using the point-slope form of a line, we can substitute the slope and the coordinates of the given point (-2,4) to find the equation of the parallel line: y - y1 = m(x - x1). This gives us: y - 4 = -3/2(x - (-2)), which simplifies to y - 4 = -3/2(x + 2).
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Add.
3/4 + 7/10
Write your answer as a fraction in simplest form.
To add the fractions 3/4 and 7/10, you need to find a common denominator, which in this case is 20. Both fractions are then converted to have this common denominator and added together to get the result 29/20.
Explanation:The subject of your question is fraction addition. To add 3/4 and 7/10, first we need to find a common denominator. In this case, the common denominator is 20. Change 3/4 into a fraction with 20 as a denominator, you get 15/20. Change 7/10 to a fraction with denominator 20, you get 14/20. Add these two fractions together, we get (15+14)/20, which simplifies to 29/20.
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Help?
Which of the following equations represents a parabola that opens up and has a vertex with a positive x-value?
A. y=-x^2+8x-15
B. y=2x^2-16x+38
C. y=2x^2+16x+24
D. y=-4x^2-14x-12
56 is what percentage of 112
What number is halfway between 8.270 and 8.280?
6, 1, -4, -9,...
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Is this sequence arithmetic or geometric? What's the common difference or common ratio? What is the recursive formula?