Derivatives in Context: Motion and Related Rates: AP Calculus AB Math Worksheet – PDF Download

AP Calculus Contextual Applications Worksheet: problems 1–5

Problem 1: Interpret a rate with units

Maya’s hot cocoa has temperature \(T(t)\) in °F, \(t\) minutes after it is poured. She finds \(T^{\prime}(5)=-3.2\). Interpret this value with units.

Problem 2: Velocity and acceleration functions

A cart moves along a track with position \(s(t)=t^{2}-6t+5\) feet at time \(t\) seconds. Find \(v(t)\) and \(a(t)\).

Problem 3: When is the cart at rest?

For the cart in the previous problem, find the velocity at \(t=4\) and the time when it is momentarily at rest. Then find its position then.

Problem 4: Where the particle turns around

A particle moves on a line with \(s(t)=t^{3}-6t^{2}+9t\) for \(0\le t\le 5\). Find the times when it is at rest and its positions then.

Problem 5: Direction of motion

For the particle in the previous problem, the graphs of \(s(t)\) (blue) and \(v(t)\) (green) are shown. On which intervals does the particle move right, and on which does it move left?

AP Calculus Contextual Applications Worksheet: problems 6–10

Problem 6: Speeding up or slowing down

For \(s(t)=t^{3}-6t^{2}+9t\) with \(0< t< 5\), find \(a(t)\) and the intervals on which the speed is increasing.

Problem 7: Total distance traveled

For \(s(t)=t^{3}-9t^{2}+24t\) (feet, seconds) find the total distance traveled from \(t=0\) to \(t=6\).

Problem 8: Linearization of a cube

Write the local linearization of \(f(x)=x^{3}\) at \(x=2\) and use it to approximate \(f(2.1)\).

Problem 9: Estimate a square root

Approximate \(\sqrt{50}\) using the tangent line to \(f(x)=\sqrt{x}\) at \(x=49\). Is the estimate too large or too small? The graph near \(x=25\) shows the same shape.

Problem 10: Estimate a logarithm

Use the tangent line to \(f(x)=\ln x\) at \(x=1\) to approximate \(\ln(1.05)\).

AP Calculus Contextual Applications Worksheet: problems 11–15

Problem 11: Estimate a cube root

Approximate \(\sqrt[3]{8.3}\) with a local linearization of \(f(x)=x^{1/3}\) at \(x=8\).

Problem 12: L’Hôpital with sine

Evaluate \(\lim_{x\to 0}\frac{\sin(3x)}{x}\).

Problem 13: L’Hôpital with a quartic

Find the value of \(\lim_{x\to 1}\frac{x^{4}-1}{x^{3}-1}\) by differentiating the numerator and the denominator.

Problem 14: Apply L’Hôpital twice

Evaluate \(\lim_{x\to 0}\frac{e^{2x}-1-2x}{x^{2}}\).

Problem 15: Logarithm over a root

Determine whether \(\lim_{x\to\infty}\frac{\ln x}{\sqrt{x}}\) exists and find its value.

AP Calculus Contextual Applications Worksheet: problems 16–20

Problem 16: Polynomial over exponential

Evaluate \(\lim_{x\to\infty}\frac{x^{3}}{e^{2x}}\).

Problem 17: Cosine and a square

Evaluate \(\lim_{x\to 0}\frac{1-\cos 3x}{x^{2}}\) and name the form that allows L’Hôpital’s Rule.

Problem 18: Expanding ripple

A pebble creates a circular ripple whose radius grows at 2 ft/s. How fast is the enclosed area increasing when the radius is 10 ft?

Problem 19: Sliding ladder

A 10-foot ladder leans on a wall. Its base is 6 ft from the wall and slides away at 1.5 ft/s. How fast is the top sliding down the wall?

Problem 20: Inflating a ball

Air is pumped into a spherical ball at 30 in³/s. How fast is the radius increasing when the radius is 3 inches? (Use \(V=\frac{4}{3}\pi r^{3}\).)

AP Calculus Contextual Applications Worksheet: problems 21–25

Problem 21: Filling a conical tank

Water is poured at 5 ft³/min into a tank shaped like a cone with its vertex down. The cone is 12 ft tall with a top radius of 4 ft. How fast is the water level rising when the water is 6 ft deep? (Use \(V=\frac{1}{3}\pi r^{2}h\).)

Problem 22: Two walkers

Diego walks east at 4 mph and Aisha walks south at 3 mph, both starting from the same corner. How fast is the distance between them changing after 2 hours?

Problem 23: Shadow of a walker

A 6-foot-tall person walks away from a 15-foot lamp post at 4 ft/s. Find the speed of the tip of the shadow and the rate at which the shadow is lengthening.

Problem 24: Marginal cost

The cost in dollars to make \(x\) bracelets at a craft fair is \(C(x)=0.02x^{2}+4x+500\). Find \(C^{\prime}(100)\), interpret it, and compare it with \(C(101)-C(100)\).

Problem 25: Pool volume

The volume of water in a pool, in gallons, is \(V(t)=600+40t-t^{2}\) for \(t\) minutes. Find when the volume is changing at \(-10\) gallons per minute, and the volume then.

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