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. MarkFL

    MHB Ahmed's Calculus Parallel Resistors Q: Rate of Change Help

    Here is the question: Here is a link to the question: Calculus Parallel Resistors Question Help? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  2. S

    Rate of change: find rate length is changing at moment

    Homework Statement An equilateral triangle with side length x sits atop a rectangle with a base length of x, with the base of the triangle coinciding with the top base of the rectangle. the height of the rectangle is 7 times the length of its base. If the combined area of the figures is...
  3. A

    Derive average rate of change formula of cos

    Q. Use the addition formula cos(u+v) = cos(u)cos(v) - sin(u)sin(v) to derive the following identity for the average rate of change of the cosine function: (cos(x + h) - cos x) / h = cos x ((cos h - 1) / h) - sin x ((sin h) / h) A. cos(x+h) = cosxcosh - sinxsinh subtitute this to...
  4. M

    Direction of the fastest rate of change of a function (Solution inclu)

    Homework Statement Find all the points at which the direction of the fastest rate of change of the function f(x,y)=x^{2}+y^{2}-2x-4y is i+j Homework Equations The Attempt at a Solution The direction of the fastest change is \nabla f(x,y)=(2x-2)i+(2y-4)j, so we need to find all...
  5. S

    No problem, glad it worked out in the end!

    Homework Statement Suppose you are climbing a hill whose shape is given by the equation: z = 1400 − 0.005x^2 − 0.01y^2 where x, y, and z are measured in meters, and you are standing at a point with coordinates (120, 80, 1264). The positive x-axis points east and the positive y-axis points...
  6. D

    Finding rate of change formulas with the given information

    Homework Statement Let Q(t)=t2. Find a formula for the average rate of change (ROC) of Q over the interval [1, t] and use it to estimate the instantaneous ROC at t = 1.Homework Equations For x1 ≠ x0, the average rate of change of y with respect to x over [x0, x1] is the ratio Average ROC =...
  7. R

    Rate of change of angular acceleration?

    I am calculating the time it takes for an object to fall 90° about an axis of rotation in the x-y plane. I have managed to calculate the rate of change of angular acceleration, which will be represented by the symbol ψ. Unfortunately it is not rad/s^3, but (rad/s^2)/θ or \alpha/θ. The value of ψ...
  8. M

    Finding direction where the rate of change is fastest

    Homework Statement surface of mountain is modeled by h(x,y)=5000-0.001x^2-.004y^2. Climber is at (500,300,4390) and he drops bottle which is at (300,450,4100) What is his elevation rate of change if he heads for the bottle? In what direction should he proceed to reach z=4100 feet the...
  9. J

    Water Tank Volume Rate of Change at Specific Depth

    Homework Statement Water runs into a cylindrical tank at the rate of 0.6m^3/s. The tank has a heigh of 2.5m and a base radius of 1.2m. How fast is the water level rising when the water level is 1.6m deep. Homework Equations V=∏ x r^2 x h The Attempt at a Solution I know that...
  10. S

    How to find this rate of change?

    Homework Statement Given f(x, y, z) = 3x + 2y + 4z, find its rate of change at (1, 2, 3) on the plane y = 2. Homework Equations The Attempt at a Solution Since y is constant, I don't know if I have to find ∂f/∂x or ∂f/∂z. It's a homework problem, but I don't know if it was bad written by the...
  11. MarkFL

    MHB Neonblast342's question at Yahoo Answers regarding finding a rate of change

    Here is the question: Here is a link to the question: Calculus problem! Please help!? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  12. deep838

    Rate of change of angular velocity

    Let a particle of mass m be rotating in a circle, radius be r, axis of rotation through the center and perpendicular to the plane of motion. Now if the radius is made decrease somehow at some rate until it get's zero, how do I find the rate of change in angular velocity? Also, the angular...
  13. S

    Problem about rate of change (multivariable calculus)

    Homework Statement Let T = f(x, y, z), where dx/dt = 4, dy/dt = 4 and dz/dt = -3 Calculate dT/dt if dT/dx = 4, dT/dy = 7 and dT/dz = 9 Homework Equations dT/dt = ∂T/∂x (dx/dt) + ∂T/∂y (dy/dt) + ∂T/∂z (dz/dt) The Attempt at a Solution I tried to get an explicit formula for T. I...
  14. P

    Simple rate of change problem?

    Homework Statement The Function S(t) = 6t(t+1) describes the distance (in kilometers) that a car has traveled after a time t ( in hours), t E [0,6] Find the average velocity of the car from t= 2 to t= 2 + ∆t Homework Equations y'= f(x+h) - f(x) ------------ h...
  15. M

    Why is rate of change related to the area under a function

    I don't understand why integrals and derivatives work, and i don't understand why theyre so closely related. Let's take a function y= x^2 + 2x + 9 y' = 2x + 2 Why do the rules for taking derivatives work? Why does reducing the power of a term by 1 and adding that as a coefficient work...
  16. L

    What is the distance traveled in the nth second?

    Homework Statement A particle moves along the x-axis in such a way that its distance x cm from the origin after t t seconds is given by the formula x= 7t + 12t2. What distance does it travel in the nth second? Isn't it 7(n) + 12(n)2?
  17. L

    Calculating Displacement and Distance for Vertical Motion Problems

    Homework Statement A ball is thrown vertically upwards and its height after t s is s m where s= 25.2t - 4.9t2 Find the distance moved in the 3rd second 2. At the instant from which time is measured a particle is passing through O and traveling towards A along the straight line OA. It...
  18. M

    Rate of Change Problem with Clock Hand Position

    Homework Statement "On a certain clock, the minute hand is 8 cm long and the hour hand is 6 cm long. How fast, in cm/min, is the distance between the tips of the hands changing at 9 am?" Homework Equations The Attempt at a Solution I should be able to transform this somewhat...
  19. S

    Rate Of Change Of Current In An Inductor

    Homework Statement A coil has self inductance L of 2 Henrys, and a resistance R of 100 Ohms. A DC supply voltage E of 100 volts is applied to the coil. Show graphically the approximate rate at which the current increases at the moment of switching on (t=0) and after 10ms. Confirm your...
  20. W

    Rate of change of volume of a rectangular box

    Homework Statement How fast is the volume of a rectangular box changing when the length = 6 cm, width = 5 cm and depth = 4 cm and the length and height are increasing at a rate of 1 cm/s and width is decreasing at a rate of 2 cm/s. Homework Equations Volume of a rectangle = l*w*h The...
  21. C

    Instantaneous Rate of Change Problem

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  22. F

    Rate of change of the area of the rebgion

    Homework Statement A board 5 feet long slides down a wall. at the instant the bottom end is 4 feet from the wall, the other end is moving down the wall the rate of 2 feet per second.at that moment how fast is the area of the region between the board, the ground and the wall changing...
  23. Z

    Difference Quotient - average rate of change

    Homework Statement Original function f(x) = -x^2+5x-2 I have calculated the difference quotient as -2x-h+5 Then use difference quotient to calculate the average rate of change: for the following x=2 and x=2+h I need this to go on to the next part of the question and i wanted to ensure...
  24. M

    Rate of change of population after 2.5 hours?

    Rate of change... derivative? Homework Statement the total number of people(in millions) present in a population at a given time is given by the function: 2t(5t+9)^1/2+12 where t represents time (in hours) find the rate in change of the population after 2.5 hours... Homework...
  25. S

    Solving for Rate of Change of Speed: 128° & 6.68 m/s2

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  26. X

    Calculating Entropy Change in Niagara Falls Waterfall

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  27. R

    Rate of change of momentum. Newton 2nd law

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  28. R

    Rate of Change in Proportional Equations - Understanding and Practicing the Rule

    Homework Statement This actually part of the Friedman equation but it's the math part I'm having trouble with. I'm simplifying down to the nitty-gritty. a'/a = x a' = ax a = e^x And the lecturer said that when a rate of change is proportional to thing itself then you multiply...
  29. B

    Rate of change of quantum levels (emission spectra) at 333 cm sized atoms

    Hi there, my comparatively ignorant mind is wondering, When an atomic transmutation occurs all of the quantum levels of the new atom also change 1) what is the actual rate of this change? would a nonplayer "observer photon" passing near the suddenly different element note the quantum level...
  30. J

    Related Rates / Calculating Rate of Change

    Homework Statement A lake is polluted by water from a plant located on its shore. Ecologist determine that when the level of pollutant is x parts per million (ppm), there will be F fish in the lake. When there are 4,000 fish in the lake, the pollution is increasing at a rate of 1.4ppm/year. At...
  31. C

    Calculating the Rate of Change of a Square's Area

    Homework Statement Find the rate of change of the area of a square whose side is 8 cm long, if the side length is increasing at 2cm/minHomework Equations A(t) = xy x(0)=8 y(0)=8 dx/dt=2 dy/dt=2The Attempt at a Solution dA/dt = dx/dt * x * dy/dt * y Using the product rule... dA/dt = dx/dt * x *...
  32. J

    Calculating Rate of Change for Cones: Homework Help

    Homework Statement I have attached a link along with my working please can someone help me http://s359.photobucket.com/albums/oo40/jsmith613/?action=view&current=Math.png Homework Equations The Attempt at a Solution
  33. W

    Rate of emf decline and rate of change of internal resistance

    Hello everyone, I am trying to work on this topic as my research. i plan on using a potentiometer but i want to be able to use the resistor (across the test cell) as the load and still be able to obtain two balance points. one for emf, and another for calculating internal resistance as the cell...
  34. C

    Calculating Population Growth Rate using Derivatives

    Homework Statement A population of 500 bacteria is introduced into a culture and grows in number according to the equation P(t)=500(1+(4t/(50+t2), where t is measured in hours. Find P'(2), an interpret the results. Homework Equations The Attempt at a Solution I used the product...
  35. N

    Average Rate of Change for f(x) = 5x^2 - 8x from x = -1 to 4

    Homework Statement Consider the graph of the function f(x) = 5x^2 - 8x Determine the average rate of change as x changes from -1 to 4 The Attempt at a Solution f(4) - f(-1) / 4 - 1 The answer is not correct according to the answer sheet. :(
  36. A

    Find the maximum rate of change

    Find the maximum rate of change of the function f(x,y) = (3 y^5)/x at the point (1,2) First I took the gradient of f(x,y)=<-3y^5*x^-2,15y^4*x^-1> and took the pt <1/sqrt(5),2/sqrt(5)> Then <-3y^5*x^-2,15y^4*x^-1>*<1/sqrt(5),2/sqrt(5)> the answer i get is 171.73 I'm not sure where my error is
  37. C

    How Does the Area of a Square Change with Its Diagonal?

    Homework Statement What is the rate of change of an area of a square with respect to the length of its diagonal (r) when the square has a side length of 5 meters? Homework Equations The Attempt at a Solution Area of a square = s^2 and if Area of a square = s^2 then it would =...
  38. A

    Instantaneous rate of change [Trig]

    Homework Statement For the function f(x) = 3sin 2x, find the instantaneous rate of change at point A (\frac{\pi}{6}, f(\frac{\pi}{6})) Homework Equations iroc = instantaneous rate of change iroc= \lim_{h\to 0} \frac{f(a+h) - f(a)}{h} The Attempt at a Solution iroc = instantaneous...
  39. A

    Finding instantaneous rate of change.

    Homework Statement After you eat something that contains sugar, the pH level in our mouth changes. This can be modeled by the function L(m)=(\frac{-20.4m}{m^2 + 36}) + 6.5 where L is the pH level and m is the number of minutes that have elapsed since eating. Find the instantaneous rate of...
  40. R

    Finding the Rate of Change of Theta in a Ship Navigation Problem

    Homework Statement Ship A is traveling due West towards a lighthouse at a speed of 15km/hr. Ship B is traveling due north away from the lighthouse at a speed of 10 km/hr. Let x be the distance between ship A and the lighthouse at time t, and let y be the distance between ship B and the light...
  41. H

    Direction of maximum increase and rate of change

    Homework Statement Homework Equations .. The Attempt at a Solution We find \nabla f(1,2) at point P: the answer is (-4, 13). Now, we know that the directional derivative of f is given by: \nabla f(1, 2) |u| \cos\theta where |u| = 1 and \cos\theta=1 (since the direction is maximized at...
  42. T

    Rate of Change of product of three functions

    Homework Statement Find the rate of change of the product f(x)g(x)h(x) with respect to x at x =1 given that f(1) = 0, g(1) = 2, h(1) = -2 f'(1) = 1, g'(1) = -1, h'(1) = 0 Homework Equations f(x), g(x), h(x) (they are not provided) The Attempt at a Solution I decided to find an...
  43. mishima

    Rate of change of separation distance

    Homework Statement This is from Calculus for the Practical Man, by J.E. Thompson, 1962 edition. "Two automobiles are moving along straight level roads which cross at an angle of 60 degrees, one approaching the crossing at 25 mph, and the other leaving at 30 mph on the same side. How fast...
  44. D

    Directional Derivatives: the maximum rate of change of f.

    The maximum rate of change of f at the given point and the direction in which it occurs. f(x,y)=(y^2)/x, (2,4) Answer: 4√2, <-1,1> _________________________ For this problem, since I couldn't find the upside down triangle, I am going to use delta to represent the gradient of a function...
  45. S

    How can you find the rate of change of momentum?

    Hi guys. Need a little help with an IB paper(Just two questions!) and would really appreciate your help. Homework Statement A bus is traveling at a constant speed of 6.2m/s along a section of road that is inclined at an angle of 6.0 degrees to the horizontal. State the value of the...
  46. J

    Finding thickness of paper based on rate of change of radius and constant speed

    I need find the thickness of a piece of paper based on the known linear speed of the paper coming off the roll and the change in diameter of the roll. I have the ability to see the linear speed. It is consistant but changes from roll to roll. I know the starting radius, instantaneous radius...
  47. 3

    Rate of change (surface area/radius)

    Homework Statement I am 99% positive I did a) correctly, as the answer is correct according to the answer in the back of my textbook. However, I cannot do b). The question: A spherical balloon is being inflated. Find the rate of change of the surface area with respect to the radius when...
  48. A

    Power is the measure of rate of change of energy in any circuit,then

    power is the measure of rate of change of energy in any circuit,then how the energy is changing in any circuit or how the change in energy happens in circuit? thanks.
  49. H

    What is the derivative of xy with respect to t, when x=8 and dx/dt=10?

    Hello everyone I have a two question my first one is differentiate with respect to t y=x^2-3x find dy/dt when x=3 and dx/dt=2 This is what I did dy/dt=2x dx/dt (I am stuck right here) my second one is xy=4 find dy/dt when x=8 dx/dt=10
  50. 1

    Rate of change of fish population.

    Homework Statement The population of a species of fish is given by the equation below. Where t is time in years. What is the rate of change of the fish population after 5 years? Homework Equations P(t)= 1.2 x 10^8(1+0.02t)^ -2 where t is time in years The Attempt at a Solution...
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