Get Expert Help and Answers to Your Rate of Change Questions

How to find instantaneous velocity from a position vs. time graph?

How to find instantaneous velocity in calculus?

By what factor does the capacitance of a metal sphere increase if it's volume is trippled?

What is the remainder when 5^79 is divided by 7?

What is rate constant?

Acceleration is defined as the rate of change of_

How to find average rate of change
How would I find the average rate of change over 8 minutes, of a person that runs at a rate of $v(t)=x\sin <!--\; \; -->(x^2-<!--\; -\; -->7x)$ ft/min? I missed when this was taught and I have no clue on how to do it. Help is greatly appreciated.

For which value of x is the average rate of change equal to the instantaneous rate of change?
The average rate of change for $f(x)=x^2+4x-<!--\; -\; -->6$ on the interval [1,3] is 8.
I am not interested in final answer but more how to get there. I am going through calculus right now and already know about derivatives and rate of change. My problem is how to get this word problem into math language and try to solve it.
Inst. rate of change is derivative when lim approaches 0 average $f(x+h)-<!--\; -\; -->f(x)$ divided by h.

How are the average rate of change and the instantaneous rate of change related for ƒ(x) = 2x + 5?
How are the average rate of change and the instantaneous rate of change related for ƒ(x) = 2x + 5 ?
Should I figure out what is similar between them to solve this question? I don't understand how they correlate

Calculating Rate of Change
At the point (0,1,2) in which direction does the function $f(x,y,z)=x{y}^{2}z$ increase most rapidly? What is the rate of change of $f$ in this direction? At the point (1,1,0), what is the derivative of $f$ in the direction of the vector $2\stackrel{^<!--\; ^\; -->}{i}+3\stackrel{^<!--\; ^\; -->}{j}+6\stackrel{^<!--\; ^\; -->}{k}$?
I assumed that the rate of change is the same as the gradient of the function, namely $▽<!--\; ▽\; -->f$. Calculating this gave me:
$▽<!--\; ▽\; -->f=\frac{\mathrm{\partial <!--\; \partial \; -->}(x{y}^{2}z)}{\mathrm{\partial <!--\; \partial \; -->}x}\stackrel{^<!--\; ^\; -->}{i}+\frac{\mathrm{\partial <!--\; \partial \; -->}(x{y}^{2}z)}{\mathrm{\partial <!--\; \partial \; -->}y}\stackrel{^<!--\; ^\; -->}{j}+\frac{\mathrm{\partial <!--\; \partial \; -->}(x{y}^{2}z)}{\mathrm{\partial <!--\; \partial \; -->}z}\stackrel{^<!--\; ^\; -->}{k}$
$=y^{2}z\stackrel{^<!--\; ^\; -->}{i}+2xz\stackrel{^<!--\; ^\; -->}{j}+x{y}^2\stackrel{^<!--\; ^\; -->}{k}$
Evaluating at point:
$▽<!--\; ▽\; -->f(0,1,2)=2\stackrel{^<!--\; ^\; -->}{i}$
Hence, the function increases most rapidly in the x direction.
I am uncertain of how to approach solving the third part of the question, should I evaluate the rate of change at (1,1,0) and then find the difference between that and the vector $2\stackrel{^<!--\; ^\; -->}{i}+3\stackrel{^<!--\; ^\; -->}{j}+6\stackrel{^<!--\; ^\; -->}{k}$?

Relative rate of change
Problem: Volume of a cubic box is $V=L^3$. How are the relative rates of change of V and L related?
This problem seems really simple, but I can't understand the concept of a relative rate of change. Here are my workings:
Original equation
$V=L^3$
I write it in form of differential equation
$\mathrm{\partial <!--\; \partial \; -->}V=3L^2\mathrm{\partial <!--\; \partial \; -->}L$
then I divide the 2nd line by the 1st
$\frac{\mathrm{\partial <!--\; \partial \; -->}V}{V}=\frac{3L^2\mathrm{\partial <!--\; \partial \; -->}L}{L^3}$
$\frac{\mathrm{\partial <!--\; \partial \; -->}V}{V}=3\frac{\mathrm{\partial <!--\; \partial \; -->}L}{L}$
Expression of the form $\frac{\mathrm{\partial <!--\; \partial \; -->}f(x)}{f(x)}$ is called a relative rate of change. And it can be thought of as percentage change. So as I see it by knowing the percentage change in L we can work out the respective percentage change in V, right? Wrong. I tried to make sense of it by putting values into the equation but no success. E.g. we increase L from 2 to 3 (by 50%) thus according to the last formula we should have a respective increase in V of 150% (50% times 3) which is not true ($\frac{3^3-<!--\; -\; -->2^3}{2^3}$is a 237.5% increase). Can you help me out? I'm definitely missing something either in computation or more likely in understanding the concept.

Rate of Change Questions?
If h(t) represents the height of an object above ground level at time t and h(t) is given by $h(t)=-<!--\; -\; -->16t^2+13t+1$ find the height of the object at the time when the speed is zero.
Suppose $h(t)=t^2+14t+7$. Find the instantaneous rate of change of h(t) with respect to t at t=2 .
Suppose $G(x)=6x^2+x+4$. Find a number b such that $G^{\prime }(b)=7$
Let $g(x)=2x^2+4x+1$. Find a value of c between 1 and 3 such that the average rate of change of g(x) from x=1 to x=3 is equal to the instantaneous rate of g(x) at x=c .
Let $F(s)=5s^2+3s+4$. Find a value of d greater than 0 such that the average rate of change of F(s) from 0 to d equals the instantaneous rate of change of F(s) at s=1.
Let $f(x)=x^2+x+13$. What is the value of x for which the tangent line to the graph of $y=f(x)$ is parallel to the x-axis?

Multivariable Instantaneous rate of change clarification
When you are computing the instantaneous rate of change for f(x,y) what do you take the derivative with respect to?
for example, for
f(x,y)=(sin(πx)cos(πy),yexy,x2+y3)
If I was to find the instantaneous rate of change for all 3 of these functions going through (1,2) with the velocity vector (3,-2) would I just take ddx of all of the functions at (3,-2)?

Derivative Instantaneous rate of change
Question:
The cumulative ticket sales for the 10 days preceding a popular concert is given by
$S(x)=4x^2+50x+50001\le <!--\; \le \; -->x\le <!--\; \le \; -->10.$
Find the instantaneous rate of change in S(x) when x=3.

the rate of change of pressure
The atmospheric pressure at sea level is approximately 14.7 psi. The pressure changes as altitude increases and can be calculated using the relation $P(h)=14.7\cdot <!--\; \cdot \; -->e^{-<!--\; -\; -->0.21h}$, where P is pressure in psi and h is altitude above sea level in miles.
Find an expression for the rate of change of pressure as altitude changes and estimate the rate of change in Denver, CO (h=1 mile).

Derivatives-Problem of rate of change
I'm studying applications of derivatives and I have the following problem about rate of change:
A raindrop falls over a surface and generated waves of circular form such that the rate of change of the radius of the wave its 636.36 ft/s and expands along 1,1s. Find the area of the circle of maximum radius.
My approach: let's call r the radius then the hypothesis means $v=\frac{dr}{dt}=636.36ft/s$. By physics we have $r=vt$ then $r=636.36\ast <!--\; \ast \; -->1,1=700ft$ is the maximum radius. Then $A=\mathrm{\pi <!--\; \pi \; -->}\ast <!--\; \ast \; -->r^2=\mathrm{\pi <!--\; \pi \; -->}\ast <!--\; \ast \; -->(700{)}^2=1539380.40f{t}^2$
Is my solution right?

Rate of Change Questions?
I have this equation for a rate of change problem below
$s=t^4-4t^3-20t^2+20t,t\ge <!--\; \ge \; -->0$
The question asks me
At what time does the particle have a velocity of 20 m/s?
How do i solve this? Basically what steps do I take to find at what time the particle has that velocity?

Cylindrical tank rate of change
Water is pouring into a cylinder with a radius of 5m and height of 20m at a rate of 3 cubic metres a minute. Find the rate of change of height when the tank is half full.
Now the Volume V = $\pi r^{2}h$ and I can determine the rate of change in Volume is $dV/dt=\pi r^{2}dh/dt$ and the rate of change of height is $dh/dt=1/\pi r^2\times <!--\; \times \; -->dV/dt$
Using that formula I can determine that the water is rising at a rate of $3/25\pi$ m/min. But I cannot seem to figure out how to factor in height so that it is half full. Or is this wrong?

Average rate of change?
How would I figure the following problem out?
Find the average rate of change of $g(x)=x^2+3x+7$ from x=5 to x=9My thought is that I would plug in 5 and 9 for the x values to get the y values. And the use the slope formula $\frac{y_2-<!--\; -\; -->y_1}{x_2-<!--\; -\; -->x_1}$

Determining the average rate of change?
Yeah, need help solving this one...
A function is given below. Determine the average rate of change of the function between x = 1 and x = 7.
g(x)= 3 + 1/2x