Differential Calculus, or Differentiation
If we have a function of one variable, ie of the form [tex]y=f(x)[/tex], then in its most basic form differentiation is the study of how a small change in one variable [tex]x[/tex] affects the other variable [tex]y[/tex].
As an real life example, consider the average speed of a moving car:
average speed = distance travelled/ time taken
Obviously, this is an average by definition, but if there existed a formal mathematical link between distance and time, could we build a function that would tell us the instantaneous velocity at every given moment? The study of differential calculus gives strategies for calculating the ratio of a little change in distance to a small change in time, and then calculating the real instantaneous speed by making the small change infinitely small.
Similarly if we wanted to find the gradient of the tangent to a curve at some particular point [tex]A[/tex] we would estimate the gradient by using a chord to a nearby point [tex]B[/tex]. As we move this nearby point [tex]B[/tex] closer to the tangent point [tex]A[/tex] the slope of the chord approaches the slope of the tangent with more and more accuracy. Again differential calculus provides techniques for us to make the point [tex]B[/tex] infinitesimally close to the point [tex]A[/tex] o that we can calculate the actual gradient of the tangent.
Integral Calculus, or Integration
Suppose we wanted to calculate the area under a curve, [tex]y=f(x)[/tex], bounded the [tex]x[/tex] =axis, and two points [tex]a[/tex] and [tex]b.[/tex] We could start by splitting the interval [tex][a,b][/tex] into [tex]n[/tex] regular strips, and estimating the area under the curve using trapezia (this is the essence of the trapezium rule which provides an estimate of such an area). If we increase [tex]n[/tex] then generally we would hope for a better approximation. The study of integration provides techniques for us to take an infinitely large number of infinitesimally small strips to gain an exact solution.
The Fundamental Theorem of Calculus
Given the above two notions, it would appear that there is no connection between them at first., The Fundamental Theorem of Calculus, on the other hand, is a theorem that connects the rate of change of the area function (which determines the area under a curve) to the function itself. In other words, the area function's derivative equals the function itself.
Visual for Fundamental Theorem of Calculus for integrals:
[tex]\int\limits^b_af {(x)} \, dx =F(b)-F(a).[/tex]
where F is an antiderivative of [tex]f[/tex]
Physics, Chemistry, all engineering sciences, statistics, economics, finance, biology, computer science, linguistics, to name but a few, are all areas that would be a desert without the use of calculus.
Leibnitz and Newton worked to define the velocity of a planet moving on a curved trajectory. That was not possible without calculus, and both had to invent differential calculus. Differential calculus allows to compare quantities along a curve, and thus their time rate of change.
All of classical physics can be summarized in this operation. Given second derivative (which is Force/mass), find the position as a function of time. This process is called integration. Half of calculus is made with integration, the other half with derivation. All of classical physics rests on these two parts of the calculus.
Quantum mechanics, quantum field theory, electromagnetism, fluid mechanics all use integration and derivation and much more. I rest my case. I hope this helps you gauge the place that calculus occupies in science.
Answer:
so what i think is that *friendshipStep-by-step explanation:
a function is graphed on coordinate plane. what is the value of the function when x = -4
Answer:
no graph un asnwerable
Step-by-step explanation:
What is the Slope-Intercept form of the equation of a line that passes through the point (5,-3) and is parallel to the line y = 2x +3?
Answer:
123456098765432112345678900000000000000000000000987654444443221111111111122222233333333333334444444444455555555555556666666666666667777777777888888888889999999000
Step-by-step explanation:
what is the answer ?
The angle B will be equal to ∠B=53.13 degrees.
What is the triangle?Triangle is a shape made of three sides in a two-dimensional plane. the sum of the three angles is 180 degrees.
It is given that:-
Three sides of the triangle are:- a= 3, b=4 and c =5
The angle B will be calculated by the following formula;-
[tex]\angle B=Cos^{-1}\dfrac{a^2+c^2-b^2}{2ac}\\\\\\\angle B=Cos^{-1}\dfrac{3^2+5^2-4^2}{2\times 3\times 5}\\\\\\\angle B=Cos^{-1}(0.6)\\[/tex]
∠B=53.13 degrees
Hence the angle B will be equal to ∠B=53.13 degrees.
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Mala walks her dog 3 times a day.
Each walk 3/4 ismile
Which expression tells how many miles Mala walks her dog each day?
Answer:
3/4 x 3 (top right)
Step-by-step explanation:
one you selected is incorrect
can someone help me with this?
Answer:
36 ft.
Step-by-step explanation:
The given perimeter of triangle EDF is 12, in which the hypotenuse is 5. This means that it is a 3-4-5 triangle, meaning that ED = 3, DF = 4, EF = 5.
It is given that BC (the other triangle's hypotenuse) is 15. This means that the left triangle is to the scale factor of 3 of triangle EDF. Therefore, multiply 3 to all sides measurements, and then add all the sides to get the perimeter.
Hypotenuse: 5 x 3 = 15
Leg 1: 4 x 3 = 12
Leg 2: 3 x 3 = 9
Perimeter: 15 + 12 + 9 = 36
36 ft. is your answer.
PLEASE HELP ILL MARK BRAINEST!!
Answer:
-3,-2
Step-by-step explanation:
Answer:
-3,-2
Step-by-step explanation:
if it goes down in numbers but increases in x that would mean the value would change by that much (-3,-2).
Find the value of y when x=-2
y=5x+6
y=
Answer:
Soln;
From the given question, We know that:
x= -2
So,
y= 5x+6
= 5*(-2)+6
= -4 Ans
Therefore, The value of y is -4.find the surface area of the composite figure. 5 cm 6 cm 4 cm 20 cm 12 cm R6 cm 5 cm 4 cm
help pls fast
Answer:
644 cm^2
Step-by-step explanation:
20*5*2 = 200
6*5*2 = 60
20*6 = 120
6*8 = 48
12*4*2 = 96
6*4*2 = 48
12*6 = 72
Which of the following statements best describes the lines?
4 4 cm 2 cm 4 cm 4 cm
What is the area of the polygon?
Answer:
512
Step-by-step explanation:
or is the 4 4 a fraction? if it was a fraction, it is 32 bc 4 4 = 1.
What is the distance between points L(-1,3) and 2(4, -9)
HELP ASAP
Answer:
13 units
Step-by-step explanation:
calculate the distance d using the distance formula
d = [tex]\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2 }[/tex]
with (x₁, y₁ ) = (- 1, 3 ) and (x₂, y₂ ) = (4, - 9 )
d = [tex]\sqrt{(4-(-1))^2+(-9-3)^2}[/tex]
= [tex]\sqrt{(4+1)^2+(-12)^2}[/tex]
= [tex]\sqrt{5^2+144}[/tex]
= [tex]\sqrt{25+144}[/tex]
= [tex]\sqrt{169}[/tex]
= 13
Step-by-step explanation:
the other answer is correct.
just a few words about the why :
when you look at 2 points on the coordinate grid, then the distance between them creates a right-angled triangle.
the direct distance is the Hypotenuse (the baseline opposite of the 90° angle).
and the x-coordinate difference and the y-coordinate difference are the legs of that triangle.
and then we can use Pythagoras
c² = a² + b²
c = sqrt(a² + b²)
with c being the Hypotenuse.
and you see, that is exactly the so-called distance formula the other answer used.
sanjeev ate 1/6 of cake he gave to his frend the remainder 1/5 how much did he keep
Step-by-step explanation:
[tex] = 1 - ( \frac{1}{6} + \frac{1}{5} )[/tex]
[tex] = 1 - ( \frac{5}{30} + \frac{6}{30} )[/tex]
[tex] = 1 - \frac{11}{30} [/tex]
[tex] = \frac{30}{30} - \frac{11}{30} [/tex]
[tex] = \frac{19}{30} [/tex]
So, he keeps [tex] \frac{19}{30}[/tex]
Answer:
Sanjeev kept 19/30 of his cake.
Step-by-step explanation:
So, if Sanjeev ate 1/6 of the cake there will be 5/6 of the cake left over.
He gave 1/5 of this 5/6 to his friend.
He did not give 1/5 of the entire cake, he gave 1/5 of the remainder of his cake, which was 5/6.
To solve this, we must first convert both fractions to the same denominator.
1/5 and 5/6 becomes 6/30 and 25/30
He gave away 6/30 of this 25/30
It is now clear that we must take away how much he gave away from the remainder of the cake.
25/30 subtract 6/30 becomes 19/30
We cannot simplify this.
Therefore, Sanjeev kept 19/30 of his cake.
Please help! Give detailed answer.
Answer:
C
Step-by-step explanation:
Pythagorean theorem
√25²-20²=15
Answer:
C
Step-by-step explanation:
the figure is a rectangle with opposite equal sides
Consider the upper right triangle
using Pythagoras' identity
p² + 20² = 25²
p² + 400 = 625 ( subtract 400 from both sides )
p² = 225 ( take square root of both sides )
p = [tex]\sqrt{225}[/tex] = 15
Please answer this qjusetion. :)
The answer to number 1 is 20%.
The answer to number 2 is the top left box. She charges $100 for each one hour lesson.
Which of the following numbers is irrational?
a fraction with numerator negative 15 and denominator 4, a fraction with numerator negative 7 and denominator 9, square root of 4, π
a fraction with numerator negative 15 and denominator 4
a fraction with numerator negative 7 and denominator 9
square root of 4
π
Answer:
Pi is an irrational number
Answer:
Square root of 4π
An irrational number is never-ending or a fraction, and pi is almost never-ending, making it the only irrational number here since it cannot be written as a fraction.
Which statement is false?
→
+7
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
-3-5
7<3
-5<-3
3<7
Answer:
I don't even have to look at it It's 17 final answer
Answer:
7 < 3 is false.
Explanation:
a) -3 > -5 is a true statement
b) 7 < 3 is a false statement
c) -5 < -3 is a true statement
d) 3 < 7 is a true statement
19. The results of a board game are shown in the table below.
Player
Scores
Joseph
500
Sarah
-1,500
Malachi
-2,200
Tyson
400
Shaun
-600
Summer
-300
I
What is the total of the players' scores?
Find the circumference of a circle that has a 25-inch radius. Use 3.14 to approximate π.
PLEASE HELP ILL GIVE BRAINLIEST
[tex] \huge \tt \color{pink}{A}\color{blue}{n}\color{red}{s}\color{green}{w}\color{grey}{e}\color{purple}{r }[/tex]
[tex] \large\underline{ \boxed{ \sf{✰\:Important\: point's }}}[/tex]
➣ Circle:- A two-dimensional geometric figure, a line, consisting of the set of all those points in a plane that are equally distant from a given point (center).➣ Circumference:- The line that bounds a circle or other two-dimensional figure[tex]{ \rule{70mm}{2.9pt}}[/tex]
★ also circumference of a circle is also known as it's perimeter
➣To find circumference of circle when use formula[tex]{ \boxed{✜\underline{ \boxed{ \sf{\: Circumference \: of \: circle= \: 2\pi \: r}}}✜}}[/tex]
➢ r = radius (25inch.)➢ value of π is 3.14[tex]{ \rule{70mm}{2.9pt}}[/tex]
[tex] \large\underline{ \boxed{ \sf{✰\: Let's\:solve }}}[/tex]
[tex]\sf{➛ \: Circumference \: of \: circle = 2 \times 3.14 \times 25} \\ \sf{➛ \: Circumference \: of \: circle =2 \times 78.5} \\ \sf{➛ \: Circumference \: of \: circle =157inch}[/tex]
★ Hence circumference of circle=157inch
[tex] \large\underline{ \boxed{ \sf{➪\:157inch. }}}[/tex]
[tex]{ \rule{70mm}{2.9pt}}[/tex]
Hope it helps !
What is the slope of the line that passes through the points (-9,-5) and
(-1,-9)? Write your answer in simplest form.
Answer:
slope= (y2-y1)/x2-x1
Step-by-step explanation:
(-5-(-9))/(-9-(-1))
=-1/2
[tex]\large{\underline{\underline{\pmb{\sf{\color{yellow}{Answer:}}}}}}[/tex]
-1/2
Step-by-step explanation:
To findSlope
Given[tex] \sf let \: ( - 9, - 5) = (x_1,y_1) \\ \\ \sf ( - 1, - 9) = (x_2,y_2)[/tex]
Solution[tex] \sf Slope(m) = \frac{y_2 - y_1}{x_2 - x_1} [/tex]
Substituting the values
[tex] \sf \implies m = \frac{ - 9 - ( - 5)}{ - 1 - ( - 9)} \\ \\ \\ \sf \implies m = \frac{ - 9 + 5}{ - 1 + 9} \\ \\ \\ \sf \implies m = \frac{ - 4}{8} \\ \\ \\ \sf \implies m = - \frac{1}{2} [/tex]
what is the volume of the pyramid? The base is 2cm and the height is 3.75
I am going to assume this is a square pyramid and the base is 4 cm², in other words, 2 * 2 = 4
Answer: 5 cm³
Step-by-step explanation:
We will use the formula for a pyramid and solve.
[tex]\displaystyle V=\frac{lwh}{3}[/tex]
[tex]\displaystyle V=\frac{(2)(2)(3.75)}{3}[/tex]
V = 5 cm³
please give answer asap i’ll give brainlist to correct answer
Answer:
I believe the answer is; The median is 4.5 inches.
let me know if this is right for you! :)
Step-by-step explanation:
what is 10x+15 if x is 10
The answer would be 115
Step-by-step explanation:
please please answer (-2)(-4)(-1)(-7)(1) = and explain
Answer:
56
Step-by-step explanation:
(-2)(-4)(-1)(-7)(1) =
(8)(-1)(-7)(1) =
(-8)(-7)(1) =
(56)(1) = 56
hope this helps
have a good day
What are the coefficients in the polynomial 4x2 + 3x-3?
•4, -3, -3
•4, 3
•4, 3, 3
•-4, -3
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
[tex]\diamond\large\blue\textsf{\textbf{\underline{\underline{Question:-}}}}[/tex]
What are the coefficients in the polynomial 4x²+3x-3?
[tex]\diamond\large\blue\textsf{\textbf{\underline{\underline{Answer and How to solve:-}}}}\diamond[/tex]
Coefficient:- A number before a variable
Coefficients in this polynomial:-
4 (the number before x²) and 3 (the number before x)
So we conclude that the right option is:-
[tex]\bigcirc\!\!\!\!\!\large\checkmark[/tex] 4, 3 (Option B)
Good luck.- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
You've got three circles in line with eachother. Radius = 4cm
Calculate the shaded area and give answer in pi.
The shaded area of the circles that are given will be 32π.
How to calculate area?It should be noted that the area of a circle is calculated by using:
= πr²
where r = 4
In this case, we have two shaded circles. The area will be:
= 2(πr²)
= 2(π × 4²)
= 2(16π)
= 32π
In conclusion, the area is 32π.
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Need help asap please explain
The sum of the first six terms of the geometric sequence 1, 2, 4, 8, ... is equal to the sum of
the first six terms of an arithmetic sequence that also has 1 as its first term. What is the
fifth term of the arithmetic sequence?
Answer:
Step-by-step explanation:
sum of first six terms of given G.P.=1+2+4+8+16+32=63
let d be c.d. of A.P.
first six terms are 1,1+d,1+2d,1+3d,1+4d,1+5d
1+1+d+1+2d+1+3d+1+4d+1+5d=63
6+15d=63
15d=63-6=57
divide by 3
5d=19
d=19/5=3.8
fifth term=1+4d=1+3.8*4=1+15.2=16.2
Alana claims that not all 4-sided polygons with 2 pairs of equal sides are parallelograms. is alana correct? use terms from the word lis in your answer.
Step-by-step explanation:
Alana is correct.
example : a kite
2 pairs of equally long sides, but none is parallel.
Juanita has 2 rolls of film with 36 exposures and 3 rolls of film with 24 exposures. How many photos can she take with this film? She can take ____ pictures
Answer:
65 pictures because 24+36+3+2
Consider the diagram below:
What is the relationship between the two highlighted angles?
They are corresponding angles.
They are consecutive interior angles
They are vertical angles.
They are alternate interior angles.
Answer:
The are consecutive interior angles.
A fair spinner has 12 equal sections: 4 red, 3 blue and 5 green. It is spun twice. What is the probability of getting the same colour twice?
Answer:
25/72
Step-by-step explanation:
Total outcomes :
(5 + 4 + 3)²(12)²144 outcomesP (red twice) :
16/1441/9P (blue twice) :
9/1441/16P (green twice) :
25/1445/12Adding :
1/9 + 1/16 + 5/1216/144 + 9/144 + 25/14450/14425/72