AP Calculus BC Unit 4: Contextual Applications of Differentiation

Find the equation of the normal line to $y=x^2-3x$ at the point $(1,-2)$.

A spherical balloon is inflated so its radius increases at $0.5$ cm/min. How fast is the volume increasing when $r=3$ cm?

A particle's position is $s(t)=2\sin (\pi t)$ for $t\ge 0$. What is the particle's acceleration at $t=1/2$?

Use L'Hôpital's Rule to evaluate ${\lim }_{x\to 1}\frac{x^3-3x+2}{x^2-2x+1}$.

A particle moves along the $x$-axis so that its position at time $t\ge 0$ is $x(t)=t^3-6t^2+9t$. At what times is the particle at rest?

A particle's position is $s(t)=2\sin (\pi t)$ for $t\ge 0$. What is the particle's acceleration at $t=1$?

Find the linearization $L(x)$ of $f(x)=\sqrt{x}$ at $a=9$, and use it to approximate $\sqrt{9.4}$.

A point moves along the curve $y=x^2$. When $x=3$ the $x$-coordinate increases at $2$ units/sec. How fast is the distance from the origin increasing?

Two cars approach an intersection; one travels north at $30$ mph and the other travels east at $40$ mph. How fast is the distance between them decreasing when they are $5$ mi and $12$ mi from the intersection, respectively?

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

Evaluate ${\lim }_{x\to 0}\frac{e^x-1-x}{x^2}$ using L'Hôpital's Rule.

A point moves along the curve $y=x^2$. When $x=3$ the $x$-coordinate increases at $2$ units/sec. How fast is the distance from the origin increasing at that moment?