This ap calculus contextual applications worksheet gives 25 problems on derivatives in context: motion and related rates, ordered from easy to hard, with a separate answer key.
These 25 problems practice derivatives in context for AP Calculus AB Unit 4: interpreting rates, motion along a line, local linearization, L’Hôpital’s Rule and related rates. They go from short interpretations to multi-step problems.
The last seven problems are word problems with pictures. Include units in every answer.
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.
How to use this ap calculus contextual applications worksheet
Print this ap calculus contextual applications worksheet as a PDF, solve every problem on paper, then compare with the answer key.
See every chapter on the AP Calculus AB math page.