Derivative as Rate of Change

Water is running into an underground right circular conical reservoir, which is $10 \mathrm{m}$ deep and radius of its base is $5 \mathrm{m}.$ If the rate of change in the volume of water in the reservoir is $\frac{3}{2}\frac{\pi {\mathrm{m}}^3}{\min }$, then the rate (in $\frac{\mathrm{m}}{\min }$) at which water rises in it, when the water level is $4 \mathrm{m}$, is

If the volume of a sphere is increasing at a constant rate, then the rate at which its radius is increasing,is

In which of the following range $x$ lies such that $9x^2-12x-40$ increases more rapidly than $2x^3+7$.

At which point on the parabola $x^2=4y,$ the rate of increase of the $x$- coordinate is the same as the rate of the increase of $y$- coordinate

If there is $2\%$ error in measuring the radius of Sphere, then .......... will be the percentage error in the surface area.

If $y=7x-x^3$ and x increases at the rate of $4$ units per second, then the rate of change in slope of the curve when $x=3$, is

A point on the parabola $y^2=18x$ at which the ordinate increases at twice the rate of the abscissa is:

At which point on the parabola $x^2=4y,$ the rate of increase of the $x$- coordinate is the same as the rate of the increase of $y$- coordinate.

A point on the parabola $y^2=18x$ at which the ordinate increases at twice the rate of the abscissa, is

The distance in$$ meters covered by a body in t seconds, is given by $s=3t^2-8t+5.$

The body will stop after

A company's profits, in thousands of dollars, can be modelled by the function: $P\left(x\right)=0.08x^3-1.9x^2+12.5x$, where $x$ is the number of units sold (in millions) each week. Write down the values of $x$ for which the instantaneous rate of change is zero. Justify your answer.

A company's profits, in thousands of dollars, can be modelled by the function: $P\left(x\right)=0.08x^3-1.9x^2+12.5x$, where $x$ is the number of units sold (in millions) each week. State the values of $x$ for which the instantaneous rate of change is positive. State the values of $x$ for which the instantaneous rate of change is negative. Explain the meaning of each of these results.

A company's profits, in thousands of dollars, can be modelled by the function: $P\left(x\right)=0.08x^3-1.9x^2+12.5x$, where $x$ is the number of units sold (in millions) each week. Calculate the instantaneous rate of change at $x=3,x=8\text{ and }x=13$. Explain the meaning of these values.

The distance of a bungee jumper below his starting point can be modelled by the function $f\left(t\right)=80t^2-160t,0\le t\le 2$, where $t$ is the time in seconds. Find $f^{\text{'}}(0.5)\text{ and }{f}^{\text{'}}(1.5)$ and comment on the values obtained.

The distance of a bungee jumper below his starting point can be modelled by the function $f\left(t\right)=80t^2-160t,0\le t\le 2$, where $t$ is the time in seconds. State the quantity represented by $f^{\text{'}}\left(t\right)$.

The distance of a bungee jumper below his starting point can be modelled by the function $f\left(t\right)=80t^2-160t,0\le t\le 2$, where $t$ is the time in seconds.Find $f^{\text{'}}\left(t\right)$.

The profit, $\text{ US\$P }$,made from selling cupcakes, $c$, is modelled by the function $P=-0.056c^2+5.6c-20$. Find the rate of change of the profit. with respect to the number of cupcakes when $c=20\text{ and }c=60$ and comment your answers.

The profit, $\text{ US\$P }$,made from selling cupcakes, $c$, is modelled by the function $P=-0.056c^2+5.6c-20$. Find $\frac{dP}{dc}$.