AP Calculus AB Unit 4: Contextual Applications of Differentiation

Apply L'Hôpital's rule to evaluate $\underset{x\to 0}{\lim }\frac{e^x-1-x}{x^2}$.

A ladder 10 ft long leans against a vertical wall. The base slides away from the wall at $2$ ft/s. How fast is the top of the ladder sliding down when the base is $6$ ft from the wall?

A 5-foot tall person walks away from a 20-foot lamppost at $3$ ft/s. How fast is their shadow lengthening?

Use L'Hôpital's rule to find $\underset{x\to 0^{+}}{\lim }x\ln x$.

A particle moves along a number line so that its position is $s(t)=t^3-6t+2$ (feet, $t$ in seconds). What is the velocity at $t=2$?

The position of a particle is $s(t)=t^3-9t^2+24t$ for $t\ge 0$. At what time(s) is the particle at rest?

The radius of a circle is increasing at a rate of $3$ cm/s. At what rate is the area increasing when the radius is $5$ cm?

A spherical balloon is being inflated so that its volume increases at $100$ cm³/s. How fast is the radius increasing when the radius is $5$ cm? (Recall $V=\frac{4}{3}\pi r^3$.)

Water drains from a conical tank (vertex down) of height 10 m and radius 4 m at $2$ m³/min. How fast is the water level dropping when $h=5$ m? (Recall $V=\frac{1}{3}\pi r^{2}h$.)

The volume of a cube is decreasing at $12$ cm³/s. How fast is the side length decreasing when the side is $2$ cm?

Using linear approximation near $x=4$, estimate $\sqrt{4.1}$.

The temperature $T$ (°F) of a cooling object satisfies $T^{\prime }(t)=-0.1(T-70)$. At time $t=0$, $T=200$°F. Using the differential equation, what is $T^{\prime }(0)$?