Rate of Change of Quantities

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 $\frac{dP}{dc}$.

A particle is moving in a straight line so that after $t$ seconds its distance $s$ (in centimetres) from a fixed point on the line is given by $s=f\left(t\right)=8t+t^3$. Find the initial velocity.

The volume of a cube is increasing at the rate of $18 {\mathrm{cm}}^3$ per second. When the edge of the cube is $\mathrm{12 }\mathrm{c}\mathrm{m,}$ then the rate in $\mathrm{c}{\mathrm{m}}^2/\mathrm{s}$ at which the surface area of the cube increases, is

Sand is pouring from a pipe at the rate of $12 {\mathrm{cm}}^3/\mathrm{s}$. The falling sand forms a cone on the ground in such a w that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand co increasing when the height is $4 \mathrm{cm}$?  [Report your answer upto the 4th decimal place]

The displacement is $\text{'}s\text{'}$ of the particle at time $\text{'}t\text{'}$ is given by $s=t^3-4t^2-52$. Find its velocity and acceleration at time $t=2s$.

The surface area of a spherical balloon is increasing at the rate of $2 {\mathrm{cm}}^2/\sec$. If the volume of the balloon is increasing at $a {\mathrm{cm}}^3/\sec$, when the radius of the balloon is $6 \mathrm{cm}$, then $a$ is

The surface area of a spherical balloon is increasing at the rate of $2 {\mathrm{cm}}^2/\sec$. At what rate $\left({\mathrm{cm}}^3/\sec \right)$ the volume of the balloon is increasing ,when the radius of the balloon is $6 \mathrm{cm}?$

Sand is pouring from a pipe at the rate of $12 {\mathrm{cm}}^3/\mathrm{s}$. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is $4 \mathrm{cm}?$ [Report your answer upto the 4th decimal place]

Water is being poured at the rate of $36{\mathrm{m}}^3/\sec$ in a cylindrical vessel of base radius $3$ meters. If the rate at which water level is rising is of the form $\frac{a}{\pi }$, then the value of $a=$

A square plate is contracting at the uniform rate of $2 c{m}^2/sec$. Find the rate of decrease of its perimeter when the side of the square is $16 cm$ long

For a gas equation $\mathrm{PV}=100$, volume $\mathrm{V}=25 {\mathrm{cm}}^3$ and pressure $\mathrm{P}$ is measured in $\mathrm{dynes}/{\mathrm{cm}}^2$. Find the rate of change of pressure when the volume is increasing at the rate of $0.25 c{m}^3/sec$

A ladder of $5 \mathrm{m}$ long rest with one end against a vertical wall of height $3 \mathrm{m}$ and the other end on the lower ground. If its top slides down at the rate of $10 cm/sec$, find the rate at which the foot of the ladder is sliding.