For $f(x)=3x^2+3x$, is the tangent-line estimate at $x=-3$ an over-estimate or an under-estimate of nearby values?

Let $M(t)$ be the mass of a culture after $t$ days, in grams. What does $M^{\prime }\left(4\right)=9$ mean?

A cube is heated so that each edge grows at $2$ centimeters per second. Its volume is $V=s^3$, where $s$ is the edge length. How fast is the volume changing when the edge is $3$ centimeters? Give the answer in cubic centimeters per second.

Evaluate ${\lim }_{x\to 5}\frac{x^2-11x+30}{x^2-12x+35}$ using L'Hôpital's Rule.

A circular ripple spreads so that its radius grows at $3$ centimeters per second, with $A=\pi r^2$. How fast is the area changing when the radius is $4$ centimeters? Give the answer in square centimeters per second.

A particle has position $s(t)=2t^3+3t^2-36t$ at time $t$. Is it speeding up or slowing down at $t=-2$?

A particle has position $s(t)=2t^3-3t^2$ at time $t$. Is it speeding up or slowing down at $t=-2$?

For $p(x)=x^2-x$, use the tangent line at $x=3$ to estimate $p(1)$.

A particle has position $s(t)=2t^3-6t^2$ at time $t$. Is it speeding up or slowing down at $t=4$?

Evaluate ${\lim }_{x\to 1}\frac{x^2+6x-7}{x^2+x-2}$ using L'Hôpital's Rule.

A particle has position $s(t)=2t^3-96t$ at time $t$. Is it speeding up or slowing down at $t=5$?

For $q(x)=-x^2+7x$, is the tangent-line estimate at $x=-4$ an over-estimate or an under-estimate of nearby values?

A particle has position $s(t)=2t^3-6t$ meters at time $t$ seconds. Find the acceleration at $t=1$.

A particle has position $s(t)=2t^3-9t^2-24t$ meters at time $t$ seconds. Find the acceleration at $t=-2$.

A particle has position $s(t)=2t^3-6t^2-48t$ meters at time $t$ seconds. At what times is the particle at rest?

A cube is heated so that each edge grows at $2$ centimeters per second. Its volume is $V=s^3$, where $s$ is the edge length. How fast is the volume changing when the edge is $6$ centimeters? Give the answer in cubic centimeters per second.

For $f(x)=-4x^2+2x$, find $dy$ when $x=2$ and $dx=\frac{1}{20}$.

A particle has position $s(t)=2t^3+6t^2-48t$ meters at time $t$ seconds. At what times is the particle at rest?

A cube is heated so that each edge grows at $2$ centimeters per second. Its volume is $V=s^3$, where $s$ is the edge length. How fast is the volume changing when the edge is $7$ centimeters? Give the answer in cubic centimeters per second.

For $v(x)=-3x^2+4x$, use the tangent line at $x=-2$ to estimate $v(-4)$.

A particle has position $s(t)=2t^3+12t^2$ meters at time $t$ seconds. Find the acceleration at $t=-3$.

For $q(x)=-3x^2-x$, find $dy$ when $x=-1$ and $dx=\frac{1}{20}$.

Let $P(x)$ be the profit from selling $x$ chairs, in dollars. What does $P^{\prime }\left(9\right)=-5$ mean?

A spherical balloon is inflated so that its radius grows at $2$ centimeters per second, with $V=\frac{4}{3}\pi r^3$. How fast is the volume changing when the radius is $8$ centimeters? Give the answer in cubic centimeters per second.

A particle has position $s(t)=2t^3-18t^2+48t$ meters at time $t$ seconds. Find the acceleration at $t=0$.

Evaluate ${\lim }_{x\to -3}\frac{x^2-3x-18}{x^2-4x-21}$ using L'Hôpital's Rule.

A cube is heated so that each edge grows at $4$ centimeters per second. Its volume is $V=s^3$, where $s$ is the edge length. How fast is the volume changing when the edge is $4$ centimeters? Give the answer in cubic centimeters per second.

A particle has position $s(t)=2t^3-3t^2-36t$ meters at time $t$ seconds. At what times is the particle at rest?

A cube is heated so that each edge grows at $5$ centimeters per second. Its volume is $V=s^3$, where $s$ is the edge length. How fast is the volume changing when the edge is $9$ centimeters? Give the answer in cubic centimeters per second.

A spherical balloon is inflated so that its radius grows at $2$ centimeters per second, with $V=\frac{4}{3}\pi r^3$. How fast is the volume changing when the radius is $3$ centimeters? Give the answer in cubic centimeters per second.