What is Rate of change: Definition and 279 Discussions

In mathematics, a rate is the ratio between two related quantities in different units. If the denominator of the ratio is expressed as a single unit of one of these quantities, and if it is assumed that this quantity can be changed systematically (i.e., is an independent variable), then the numerator of the ratio expresses the corresponding rate of change in the other (dependent) variable.
One common type of rate is "per unit of time", such as speed, heart rate and flux. Ratios that have a non-time denominator include exchange rates, literacy rates, and electric field (in volts per meter).
In describing the units of a rate, the word "per" is used to separate the units of the two measurements used to calculate the rate (for example a heart rate is expressed "beats per minute"). A rate defined using two numbers of the same units (such as tax rates) or counts (such as literacy rate) will result in a dimensionless quantity, which can be expressed as a percentage (for example, the global literacy rate in 1998 was 80%), fraction, or multiple.
Often rate is a synonym of rhythm or frequency, a count per second (i.e., hertz); e.g., radio frequencies, heart rates, or sample rates.

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  1. chwala

    Rate of Change: Bees in Wildflower Meadow (a-c)

    part (a) The number of Bees per Wildflower plant. part (b) ##\dfrac{dB}{dF}= \dfrac{dB}{dt} ⋅\dfrac{dt}{dF}####\dfrac{dB}{dF}=\left[\dfrac{2-3\sin 3t}{5e^{0.1t}}\right]## ##\dfrac{dB}{dF} (t=4)= 0.4839##part (c) For values of ##t>12## The number of Bees per wildflower plants reduces...
  2. C

    Why is current defined as the rate of change of charge?

    Homework Statement:: This isn't a homework question but just a theoretical questions. [mentor’s note: moved to a more appropriate forum for theoretical questions.] I know that current is defined as the rate of change of charge per unit time. i = dq/dt This makes sense for a capacitor which...
  3. D

    I Limits confusion -- What is meant by "instantaneous rate of change"?

    Suppose a particle is falling under the pull of gravity, the distance it has fallen is given by s=16t^2.Suppose we wish to find the instantaneous speed at t=1. Find the average speed between t=1 and t=1+h where h is any real number except 0. Distance traveled/Time it takes to travel the distance...
  4. K

    I Rate of change of ##L## in a rotating coordinate system

    * We've a vector ##\mathbf{A}## lying in space, changing according to some rule. * We introduce an inertial frame and find ##\left(\frac{d}{d t} \mathbf{A} \right)_{i n}## in it. * We also introduce a co located frame rotating with ##\mathbf{\omega}##. In this rotating frame I find...
  5. L

    Rate of change of acceleration

    A particle traveled in a straight line in such a way that its distance (S) from a given point on that line after time (t) was S= 20t^3 -t^4 The rate of change of acceleration at time t=2 is what value?ok, I am kind of stuck on this very simple problem. It should be as simple as taking the...
  6. ergospherical

    A Rate of Change of Bondi-Sachs Mass: Derivation Reference

    If ##(u,v)## are null coordinates and ##S(u,v)## a two-surface of constant ##u## and ##v## then the Bondi-Sachs mass ##M_{\mathrm{BS}}(u) = -\dfrac{1}{8\pi} \displaystyle{\lim_{v \rightarrow \infty}} \oint_{S(u,v)}(k-k_0) \sqrt{\sigma} d^2 \theta## satisfies (Poisson, 2007)\begin{align*}...
  7. F

    I Why is the derivative defined as the limit of the quotient?

    Hello, The derivative ##\frac {dy} {dx}## "appears" at first glance to be just the ratio of two infinitesimal quantities ##dy## and ##dx##. However, infinitesimals are not really very very small numbers even if sometimes it is useful and practical to think about them as such. Infinitesimal are...
  8. R

    When is the rate of change of Kinetic Energy maximum here?

    Because, ##F=ma=kv##, therefore, ##a=kv/m##. Clearly, the net acceleration ##A=-(g+a)##. Also, ##A=dv/dt=-(g+ \frac {kv} m )##, so cross multiplying and integrating LHS with respect to ##v## and RHS with respect to ##t## gives me: $$ v= e^{ \frac {-tk} m } * (u + \frac {gm} k) - \frac {gm} k $$...
  9. O

    I What is meant by rate of change with respect to volume?

    In physics we often come across $$\rho=\dfrac{dq}{dV}$$ Does it mean: ##(i)## ##\displaystyle \lim_{\Delta V \to 0} \dfrac{\Delta q}{\Delta V}## OR ##(ii)## ##\dfrac{\partial}{\partial z} \left( \dfrac{\partial}{\partial y} \left( \dfrac{\partial q}{\partial x} \right) \right)## What does...
  10. chucho11028

    Find the rate of change of velocity

    Hello everyone, I have a problem where I have to find the final displacement and final velocity which I have found however, I want to post few variations for that same problem which I am curious. It is more for my own knowledge > I would appreciate any help Please follow graph below with...
  11. archaic

    Dealing with Undefined Values in Calculus: A Lesson in Limits

    This is not to ask for a solution, rather I want comments on the rigor of the step. Thank you for your time! The graph shows a system of two points at ##t## and ##t+dt##, it is a bit exaggerated of course. The upper point is moving strictly in the ##x## direction and has a constant velocity...
  12. V

    Finding the derivative and rate of change

    I tried solving this question a few ways and this one logically made the most sense however I got it wrong and I am unsure of why. I first plugged in t=2 into p(t). p(2)=0.3(2)1/2+6.3 to obtain 6.724264069. I then found the derivative of D(p) which is D'(p)=-60000/p3. I plugged in...
  13. jisbon

    Rate of change (ice ball melting)

    So to begin this question, I do know that volume =4/3 pi r cubed, while the surface area) 4 pi r squared. I will like to clarify some things about the question: 1) does the first sentence means dv/dt is proportional to 4 pi r squared? 2) given the second sentence how am I able to construct an...
  14. B

    MHB Average Rate Of Change Formula

    Is this the correct symbol to use for average rate of change: \overline{\triangle}=\dfrac{f(b)-f(a)}{b-a}
  15. C

    Rate of change of area with deformation

    Could I please get a hint on how I should start this question/how I should parameterize these variables? I'm going to head to sleep as I am from the eastern time zone. I apologize ahead of time for my delayed reply.
  16. Specter

    Instantaneous rate of change and deriving a fraction

    Homework Statement Sorry for so many posts lately, hopefully this is allowed. What tangent points on ##g(x)=\frac {12} {x+1}## has an instantaneous rate of change of -3? Homework EquationsThe Attempt at a Solution [/B] I know that once I derive ##g(x)=\frac {12} {x+1}## I can set the...
  17. L

    Quick Question: Rate of Change of a Rotating Vector

    Is the equation presented (that the time-derivative of a given vector in such a scenario is equal to its angular frequency vector cross the vector itself) true in the case of a vector whose origin is not on the axis of rotation? The way I'm visualizing this, if we take such a displaced origin...
  18. bornofflame

    RC Circuit - Rate energy is dissipated in the resistor

    Homework Statement 1. A 2.01 uFcapacitor that is initially uncharged is connected in series with a 6.51 kΩ resistor and an emf source with 74.6 V and negligible internal resistance. The circuit is completed at t = 0. a) Just after the circuit is completed, what is the rate at which electrical...
  19. opus

    Rate of Change of Angle for 90 ft Wire Attachment

    Homework Statement A pole stands 75 feet tall. An angle θ is formed when wires of various lengths of ##x## feet are attached from the ground to the top of the pole. Find the rate of change of the angle ##\frac{dθ}{dx}## when a wire of length 90 ft is attached. Homework EquationsThe Attempt at...
  20. J

    Engineering Rate of change of current in an R-L Circuit

    Homework Statement Homework Equations Current cannot change suddenly through inductor unless input is impulse voltage source. The Attempt at a Solution since input is DC source we have i(0+) = i(0-) so we have di/dt at t = 0+ is 0. Is it correct? Book has mentioned answer as B which is weird...
  21. B

    Linear Algebra System of Equations/Rates Application Help

    Homework Statement Suppose that we have a system consisting of two interconnected tanks, each containing a brine solution. Tank A contains x(t) pounds of salt in 200 gallons of brine, and tank B contains y(t) pounds of salt in 300 gallons of brine. The mixture in each tank is kept uniform by...
  22. R

    I Rate of change of area under curve f(x) = f(x)

    What lead to the equality of the, rate of change of area under curve f(x) = f(x). Was it, they were just compared(OR believed to be equal) and mathematically found to be equal. Or when one was integrated or differentiated the other appeared. Also I knew, integration was being used since...
  23. CDreyer23

    Rate of Change of Magnetic Field in Square wire loop

    Homework Statement The circuit shown is in a uniform magnetic field that is into the page. The total current in the circuit is 0.20 A, flowing counterclockwise. At what rate is the magnitude of the magnetic field changing? Is it increasing or decreasing? Square wire loop with base and height...
  24. J

    Calculating Instantaneous Rate of Change for a Quadratic Function

    Homework Statement Estimate the instantaneous rate of change of the function f(x)=3x^2 + 4x at (1,7) Homework Equations ∆f(x)/∆x = f(x2)-f(x1) / x2-x1 The Attempt at a Solution I know that x=1 given the point, but to find the instantaneous rate of change I can use x=1.001 as this is a very...
  25. S

    Calculating Angle of Elevation Rate of Change for an External Elevator

    Homework Statement A building has an external elevator. The elevator is rising at a constant rate of ##2 \; \text{ms}^{-1}##. Sarah is stationary, watching the elevator from a point 30m away from the base of the elevator shaft. Let the angle of elevation of the elevator floor from Sarah's eye...
  26. Asmaa Mohammad

    The rate of change in the magnetic field linkage in a coil

    Hi, I would appreciate it if someone explains the difference between these two phases regarding a coil rotates in a uniform magnetic field. 1. The rate with which the coil intercepts the lines of magnetic field in the dynamo is maximum when the plane of the coil is parallel to the lines of the...
  27. A

    What is the rate of change of the area of a rectangle?

    Homework Statement Find the rate of change of the area of a rectangle whose area is 75cm^2. The length is 3 times the width. The rate of change of the width is 2cm/second. Homework EquationsThe Attempt at a Solution A=75 A'=? L=3x= 15 L'=6 W=x=5 W'=2 A'=L'W+LW' A'= (6)(5)+ (15)(2) A'=60cm/sec
  28. T

    Gradient and finding the direction of maximum rate of change

    Homework Statement Hi guys, it a very simple question, but it causing me a great deal of confusion. The questions are as follows: So I worked out the ans for one which I have displayed below. But what I don't understand is what they want from the second question. Because the way I see it...
  29. R

    Why is force defined as the rate of change of momentum?

    How did this definition come to Newton's mind? Force seems to be the effort we apply. So, qualitatively, more the effort, more the force. How to understand intuitively that this formula given by Newton gives a quantitative measure of Force? And, how do we know that Force does not depend on more...
  30. C

    Rate of Change of Vector in Rotating Frame

    I recognize the rate of change of a vector in an inertial frame S can be related to the rate of change of the vector in a rotating frame S0 by the equation below taken from my textbook, where Ω is the angular velocity vector. $$\Big(\frac{dQ}{dt}\Big)_{S_{0}}= \Big(\frac{dQ}{dt}\Big)_{S} +...
  31. moenste

    Rate of change of current with time, galvanometer deflection

    Homework Statement (a) What is meant by the statement that a solenoid has an inductance of 2.0 H? A 2.0 H solenoid is connected in series with a resistor, so that the total resistance is 0.50 Ω, to a 2.0 V DC supply. Sketh the graph of current against time when the crrent is switched on. What...
  32. V

    Estimated Instantaneous Rate Of Change

    Homework Statement You have to graph f(x) = x3 - 2x2 + x and find where the instantaneous rate of change is positive negative and zero. Homework EquationsThe Attempt at a Solution I factored this and found the zero's to be x= 0 and x = 1. With all this info given I graphed it and realized...
  33. Clara Chung

    Rate of change problem (differentiation)

    Homework Statement Refer to the photo, please verify my answer Homework Equations calculus The Attempt at a Solution For c, can I do it by assuming Ah=V. A(dh/dt) + h(dA/dt) = dV/dt then find dA/dt?
  34. nysnacc

    Rate of change and integrating factor

    Homework Statement Initially 5 grams of salt are dissolved in 20 liters of water. Brine with concentration of salt 2 grams of salt per lter is added at a rate of 3 liters per minute. The tank is mixed well and is drained at 3 liters a minute. How long does the process have to continue until...
  35. alaa amed

    Average rate of change of the area of the triangle?

    Homework Statement An object is moving around the unit circle with parametric equations x(t)=cos(t), y(t)=sin(t), so it's location at time t is P(t)=(cos(t),sin(t)) . Assume 0 < t < π/2. At a given time t, the tangent line to the unit circle at the position P(t) will determine a right triangle...
  36. T

    Zero-Divergence of Rate of Change of Magnetic Flux

    Let's introduce a time-varying scalar field ρ(x,y,z,t) [charge density] and vector field J(x,y,z,t) [current density] Assuming the system follows Maxwell's equations, what must both fields satisfy such that ##∇⋅(\frac{∂B}{∂t})=0## ?
  37. Jason Sylvestre

    I Gradient Vector- largest possible rate of change?

    Hello, My professor just gave us a True or False problem that states: ∇H(x,y), the gradient vector of H(x,y), gives us the largest possible rate of change of H at (x,y). Now, he said the answer is true, but it was my understanding that the gradient itself gives the direction of where the...
  38. R

    Multivariable calculus: find the rate of change

    The problem is: The temperature (in degrees Celsius) of a metal plate, located in the xy -plane, at any point (x, y ) is given by the function of two variables T(x, y ) = x sin y + y2 sin x. (a) Find the rate of change in temperature in the direction of the positive x-axis at the point (π, π)...
  39. A

    Force and rate of change of momentum

    My question is simply..if force does equate to the rate of change of momentum, then why is it not taughted as this rate rather simply a push or pull? Is it becuase really they are the same thing and it is much easy to explain/work with? Just curious Guess up until now I didn't even think of...
  40. C

    "speed rate of change" vs. "velocity rate of change"

    Homework Statement At a moment in time a particle has a velocity of v = 3 m/s î + 4 m/s ĵ, and an acceleration of a = 6 m/s2 î + 3 m/s2 ĵ. Find the rate of change of the speed of the particle, that is, find d|v|/dt. (better formatted reference...
  41. R

    MHB Help with finding Rate of change of f

    Hi! This is my first post and I'm hoping to receive so help :D If f(x)=\frac{1}{{x}^{2}+1}. The rate of change of f at a is defined as \lim_{{h}\to{0}}\frac{f(a+h)-f(a)}{h}. In case, these commands don't work properly... the question is: if f(x)=1/(x^2+1) and rate of change of f at a is defined...
  42. C

    MHB Find rate of change (Why is my answer wrong?)

    The temperature at a point (x, y) on a flat metal plate is given by T(x, y) = 62/(9 + x2 + y2) where T is measured in °C and x, y in meters. Find the rate of change of temperature with respect to distance at the point (1, 1) in the x-direction and the y-direction. My solution so far: dT/dx...
  43. Alan I

    Detector moving toward charged ring - rate of change of V

    Homework Statement A thin ring (radius r = 1.41 cm) carries a charge Q = 8.57 pC distributed uniformly along its length. The ring lies in the y-z plane, so the axis through its center is the x-axis . A small detector is moving along the positive x-axis toward the ring at velocity v = -0.543i...
  44. B

    Football Trajectory: Radius & Speed Rate of Change

    Homework Statement A football player releases a ball at 35° with initial velocity of 80 ft/sec. Determine the radius of curvature of the trajectory at times t = 1 sec and t = 2 sec, where t = 0 is the time of release from the quarterback's hand. For each case, compute the time rate of change...
  45. Taryn1

    MHB Average rate of change in the function from x1 to x2

    So this is my first question here, and I hope I'm doing it right! My question is basically this: Find the average rate of change of the function from x1 to x2. f(x) = x^2 + 12x -4 I'm also new to precalc, so please don't blame me if this is a really easy question! It doesn't seem to make...
  46. hideelo

    Expected rate of change, if at all, of nature's constants

    I am reading gravitation and spacetime by Hans Ohanian and he is discussing the possibility of the constants, such as proton mass, fine structure constant, etc, actually changing over time. He makes the claim that since the universe is ~10^10 years old the expected change should be...
  47. Amrator

    Rate of Change Using Inverse Trig Functions

    Homework Statement A spectator is standing 50 ft from the freight elevator shaft of a building which is under construction. The elevator is ascending at a constant rate of 20 ft/sec. How fast is the angle of elevation of the spectator's line of sight to the elevator increasing when the elevator...
  48. T

    Rate of Acceleration: Find Unit of Change

    if acceleration is defined as rate of change in velocity, then how do you find out the unit of rate of change of acceleration?
  49. C

    Rate of change of area in sphereical baloon, .

    Homework Statement I have a homework assignment that I find challengingA spherical baloon is being inflated at a rate of 10 cu.in/sec (i assume it's cubic inches per second) Find the rate of change of area when the baloon has radius=6Homework Equations So far I know that V=(4/3)*pi*r^3...