Rate of change Definition and 279 Threads

Homework Statement a) A large tank initially holds a mixture of 100 litres of water and 10 kg of salt. Fresh water is pumped in at a rate of 2L/min. The mixture is kept uniform by stirring and is pumped out at a rate of 2L/min. If x is the amount of salt present at any time t minutes, find the...

Homework Statement POwer P in watts, produced by an engine is P(R) = 130R / [(R+0.8)2 ] where R is resistance. What is the rate of change of power when R = 5? Homework Equations The Attempt at a Solution I do not understand how the rate of change for this is 130/ ( (R+0.8)2) -...

Suppose z=e^(xy+x-y). How fast is z changing when we move from the origin towards the point (2,1)? Would it just be 1/sqrt((2^2+1))=1/sqrt(5), i am asking this because does the z=e^(xy+x-y) get used at all?

Homework Statement Suppose that the human body dissipates a drug at a rate proportional to the amount y of drug present in the bloodstream at time t. At time t = 0 a first injection of Y0 grams is made into a body that was drug free prior to t = 0. a) Amount of drug at the end of T hours...

Does a circle have a constant rate of change of it self that defines it as a circle. I have never heard of such a thing but I am curious. It must right?? Since every circle is the same no matter the radius(the arch will change by some constant amount per radian) If that doesn't make...

Homework Statement See attachment. 2. Homework Equations /solution attempt Part (a) Well, the gradient evaluated at (1,2-1) will give the rate of change. If we want the maximum rate of change then we need the directional direction such that the unit vector \mathbf{u} is in the same...

Hey guys, I was wondering if you could just check this problem for me (I put it in a Word Document and attached it to this post). The problem, my work, and my attempted solution is included in it. Thanks!

I already posted about this but I redid the problem and got another answer: 1. Homework Statement The volume of an air mattress is 15 cubic feet. Air escapes at a rate of r(t)=0.25e^(−0.05t), where r is in cubic feet per second. Assuming the mattress is still completely inflated when the...

Homework Statement The volume of an air mattress is 15 cubic feet. Air escapes at a rate of r(t)=0.25e^(−0.05t), where r is in cubic feet per second. Assuming the mattress is still completely inflated when the valve is opened, how much air is released in the first 30 seconds?Homework Equations...

The gas tank of a parked pickup truck develops a leak. The amount, V, in liters, of gas remaining in the tank after t hours can be modeled by the function V(t)= 90(1-t/18)^2 , t is between 0 and 18. b) how fast is the gas leaking from the tank at t=12h I tried V(12) and V'(12) but still...

It is for my university course but I didn't find it worthy of the advanced physics section. Homework Statement A circular metallic disk of radius b rotates about an axis perpendicular to its centre with an angular velocity ω in a constant magnetic field, strength B, which lies in the same...

Homework Statement A 6-ft man walks away from a 15-ft lamppost at a speed of 3 ft/s. Find the rate at which his shadow is increasing in length. Homework Equations Pythagorean Theorem, Theorem of Similar Triangles (ratios of corresponding sides are equal) The Attempt at a Solution...

Homework Statement The temperature at a point (x, y, z) is given by the following equation where T is measured in °C and x, y, z in meters. T(x, y, z) = 200e-x2-3y2-9z2 Find the rate of change of temperature at the point P(2, -1, 2) in the direction towards the point (3, -3, 3). Homework...

Homework Statement A cone has the base radius of 8 cm and height of 10 cm. The height of the cone is changing at at a rate of 1cm/hour whilst radius of the base is changing with it keeping the volume constant. At what rate is the surface area of the entire cone changing at this exact...

The question: A particle in a cyclotron gaines energy q \Delta V from the alternating power supply each time it passes from one dee to the other. The time interval for each full orbit is T = \frac{2 \pi}{\omega} = \frac{2 \pi m}{q B} so the particle's average rate of increase in...

Homework Statement A 1.8 m tall student is trying to escape from the minimum security prison in Tono. She runs in a straight line towards the prison wall at a speed of 4.0 m/s. The guards shine a spotlight on the prisoner as she begins to run. The spotlight is located on the ground 30 m...

lets say there is a pond of 1,000,000 gallons of water. and the total of 10,000g of chemical is evenly dissolved in the water Fresh water starts pouring into the pond at the rate of 300 gallons/hour and the pond also releases water at the rate of 300 gallons/hour (hence no change in total...

Homework Statement Given the function f(x)= (x-2) / (x-5), determine an interval and a point where the ave. R.O.C and the instantaneous R.O.C are equal. Homework Equations IROC = [ f(x+h) - f(x) ] /h AROC = f(x2) - f(x1) / x2 - x1 The Attempt at a Solution I know that in order...

I'm not sure if this is the correct section to post in, but aside from that: I'm trying to write an expression for the rate of change of the volume of the universe based on the assumption that omega is less than 1, indicating a spherical universe. For a sphere, the volume is a function of it's...

Homework Statement For the circuit shown in Fig. 7, what is the rate of change of the current in the inductor when the current in the battery is 0.50 A? (1) 600 A/s (2) 400 A/s (3) 200 A/s (4) 800 A/s (5) 500 A/s Homework Equations i(t)=Imax(1-e-tR/L) induced emf = L (dI/dt)...

Average Rate of Change Word Problem Help!? 12. A ball is thrown vertically upward from a roof of a school that is 20 m high. The equation which gives its height above ground, in metres, after t seconds, is h(t) = - 4.9t2 + 3t + 20, a) Calculate the average rate of change of the ball between...

Homework Statement A rocket is fired vertically into the air at a rate of 6 mi/min. An observer on the ground is located 4 miles from the launching pad. When the rocket is 3 miles high, how fast is the angle of elevation between the rocket and the observer changing? Be sure to specify units...

Homework Statement A 10' ladderis leaning against a house when its base starts to slide away By the time the base is 6' fromthe house, the base is moving away at a rate of 16 ft/sec a)What is the rate of change of the height of the top of the ladder? b)At what rate is the area of...

1. Homework Statement When a condenser discharges electricity, the instantaneous rate of change of the voltage is proportional to the voltage in the condenser. Suppose you have a discharging condenser and the instantaneous rate of change of the voltage is -0.01 of the voltage (in volts per...

Homework Statement When a circular plate of metal is heated in an oven, its radius increases at a rate of 0.04 cm/min At what rate is the plate's area increasing when r is 54 cm Homework Equations The Attempt at a Solution let r be the radius dy/dr = 0.04 cm/min...

Homework Statement A trough is 15ft long and 4ft wide. Its ends are isosceles triangles with a height of 3ft. Water runs into the trough at the rate of 2.5ft^3/min. How fast is the water level rising when it is 2ft deep? Homework Equations V= .5lwh The Attempt at a Solution...

Homework Statement The equation z(e^(xy)) + z^5 + y = 4 implicitly defines z as a function z = f(x,y) near (0,2,1) (a) find df/dx and df/duy where x = 0, y = 0, and z = 1 (b) find the maximum rate of change of f at the point (0,2) Homework Equations sorry, my first post here not sure what i...

Hi, I am in calculus and am having major struggles. If someone could provide a walk through on how to answer these questions, that would be fantastic. Cheers! Let f(x)=−3x+6 if x<-3 = 15 if x > -3 Find the average rate of change of f(x) on the interval −5<x<5 . The average rate of...

I need help understanding a problem for my homework assignment. I'm not sure how to set up the problem. If anyone could help I would greatly appreciate it. Homework Statement Water is being pumped into an inverted conical tank at a constant rate. However, water is also leaking out of the...

A child of mass 22 kg swings at the end of an elastic cord. At the bottom of the swing, the child's velocity is horizontal, and the speed is 6 m/s. At this instant the cord is 3.70 m long. At this instant, what is the perpendicular component of the rate of change of the child's momentum...

An Isosceles triangle has two equal sides of length 10cm. Let x be the angle between the two equal sides. a. Express the area A of the triangle as a function of x in radians. b. Suppose that x is increasing at the rate of 10 degrees per minute. How fast is A changing at the instant x =...

Homework Statement According to Ohm's law, the voltage V, current I, and resistance R in a current are related by the equation V = IR, where the units are . . . Assume the voltage is constant V = 12V. Calculate: a) The average ROC of I with respect to R for the interval from R = 8 to R = 8.1...

Hello, I have been pondering how to plot the instantaneous rate of change of one capacitor into another through a resistive impedance R. So for example you would have C1=1 farad (100volts) C2= 10 Farad (Zero volts) R1 = 100ohms So one would have Capacitor 1 at 100 volts, discharge...

Given a convergent sequence x_n \rightarrow x and a function f, is lim_{n \rightarrow \infty} \frac{f(x_n) - f(x_{n-1})}{x_n - x_{n-1}} = f'(x) ? I believe it it is, but I haven't been able to figure out how to prove it. Does anyone know of a proof or counter-example? And probably should...

Homework Statement Find any and all points inside the interval (0,3) where the instantaneous rate of change of f equals the average rate of change of f over the interval [0,3], for the equation f(x) = 4x^2 - x^3 Homework Equations The Attempt at a Solution Not really sure how...

Imagine a boat on the water being putted towards the shore by a winch mounted a height h above the water. The winch reels in cable (shortening the hypotenuse) at constant rate u in m/s . When the boat is a distance d from the shore, find its speed and acceleration. This is an example of motion...

Homework Statement For a particular sample of monatomic gas, the rate of change of pressure with volume under isothermal conditions is -2 x 10^7 Pa m^-3. Calculate the rate of change under adiabatic conditions. Homework Equations isothermal: dP/dV = -2 x 10^7 Q = nRTln(Vf/Vi)...

I am new to Calculus. I know that the rate of change in some thing is called it's derivative or we differentiate something to find the rate of change in it. But while differentiating or better to say proving that if f(x)= sinx then f'(x)= cosx, we take f'(x)= lim h->0 [sin(x+h)- sinx]/h. Can you...

3. The volume of a tank used to provide water for animals at an animal enclosure at the local show is given by V(h) = (0.2h^3+3h)m^3 where h metres is the depth of the water in the tank at any time t seconds. Water is being added to the tank at a constant rate of 0.4 m3 per second. Find the...

Homework Statement At a given instant, a 3.8 A current flows in the wires connected to a parallel-plate capacitor. What is the rate at which the electric field is changing between the plates if the square plates are 1.76 cm on a side?Homework Equations The Attempt at a Solution I don't really...

Homework Statement A swimming pool is 5 m wide, 10 m long, 1 m deep at the shallow end, and 3 m deep at its deepest point. A cross-section is shown in the figure. If the poole is being filled at a rate of 0.1 m^3 per minute, how fast is the water level rising when the depth at the deepest...

Homework Statement I have just started my first ever calculus course, and I'm having a little trouble with a simple rate of change problem. It says: Find the average rate of change of the given function between the following pairs of x-values. The given function is f(x) = x2+ x The given...

Hi An object with temperature 26 degrees Celsius is placed in water with constant temperature of 90 degrees Celsius. If the temperature of the object rises to 70 degrees Celsius in five minutes, what will be the temperature after 10 minutes? I thought of using Newtons law of cooling...

Hi Im trying to write an equation for the question below. i did write one but am pretty sure it is wrong because I need to differentiate arctan in it and we have not been taught that yet. Could someone please point me in the right direction with writing it? Homework Statement A...

Homework Statement The concentration c of fish away from a feeding point in a lake is given by c(x,y) = 1/(x2 + y2). Find the total change of fish concentration detected by an observer riding a boat traveling with a speed u = 10 m/s straight away from the feeding point. What is the...

Is differentiation just finding the rate of change in a function? \frac{dx}{dy} (y=6x) =6? rate of change in \frac{2}{y}=\frac{2}{x}the same as :\frac{dx/dy}{dy/dx}*y=\frac{5y}{2} if and only if y=2/5 ? which equals 1 lol I think I am right :x but non- of the Wikipedia articles...

A man 1.8 m tall walks at speed of 10m/s towards a street lamp which is 7m above the ground. How fast is the length of the man's shadow decreasing? My attempt: Let the man's shadow and the lamp's length be l, and the distance be d and time be t. Given, dd/dt = 10. I am supposed to...

Homework Statement A trough in the shape of an isosceles trapezoid is 30 cm wide on the bottom, 80 cm on the top, 50 cm tall, and 10 cm long. It is being filled with water at 0.2 cm3/min. How fast is the water level rising when the water is 30 cm deep. The Attempt at a Solution Well I...

Homework Statement Consider a rectangular block of isotropic material of dimensions a, b and c, with c >> a or b. It is characterised by its elastic constants: Young's modulus E, shear modulus G and Poisson's ratio . The block of material is subjected to axial deformation along the c...

Homework Statement A conical tank (with vertex down) is 10 feet across the top and 12 feet deep. If water is flowing into the tank at a rate of 10 cubic feet per minute, find the rate of change of the depth of the water when the water is 8 feet deep. The Attempt at a Solution First I...