Applications of Calculus Cheat Sheet

A printable reference covering optimization, related rates, motion, area, volume, average value, and linear approximation for grades 11-12.

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Applications of calculus use derivatives and integrals to model change, motion, accumulation, and optimization. This cheat sheet helps students connect symbolic rules to real problem situations. It is especially useful when deciding whether a problem needs a derivative, an integral, or both. Students in grades 11 and 12 can use it as a quick reference for common setup patterns. Derivatives describe rates of change, slopes, velocity, acceleration, and maximum or minimum values. Integrals describe accumulated change, area, volume, displacement, and average value over an interval. Optimization problems usually require defining a function and solving $f'(x) = 0$ , while related rates problems usually require differentiating an equation with respect to time $t$ . Careful units, diagrams, and domain restrictions are essential in every application.

Key Facts

Critical numbers occur where $f'(x) = 0$ or where $f'(x)$ is undefined, and candidates for absolute extrema on $[a,b]$ include critical numbers and endpoints.
For optimization, write the quantity to maximize or minimize as one function, then solve $f'(x) = 0$ and check endpoints or use a derivative test.
In related rates, differentiate both sides of an equation with respect to time using implicit differentiation, such as $\frac{d}{dt}(x^2 + y^2) = 2x\frac{dx}{dt} + 2y\frac{dy}{dt}$ .
For motion along a line, velocity is $v(t) = s'(t)$ and acceleration is $a(t) = v'(t) = s''(t)$ .
Displacement over $[a,b]$ is $\int_a^b v(t)\,dt$ , while total distance traveled is $\int_a^b |v(t)|\,dt$ .
The area between curves $y = f(x)$ and $y = g(x)$ on $[a,b]$ is $A = \int_a^b |f(x) - g(x)|\,dx$ .
The average value of a continuous function on $[a,b]$ is $f_{\text{avg}} = \frac{1}{b-a}\int_a^b f(x)\,dx$ .
A linear approximation near $x = a$ is $L(x) = f(a) + f'(a)(x-a)$ .

Vocabulary

Critical number: A critical number is a value $c$ in the domain of $f$ where $f'(c) = 0$ or $f'(c)$ is undefined.
Absolute extremum: An absolute extremum is the greatest or least value of a function on its entire given domain or interval.
Related rates: Related rates problems use derivatives with respect to time to connect changing quantities in the same situation.
Accumulation: Accumulation is the total amount built up over an interval and is often represented by an integral such as $\int_a^b r(t)\,dt$ .
Average value: The average value of $f$ on $[a,b]$ is the constant height $\frac{1}{b-a}\int_a^b f(x)\,dx$ with the same signed area.
Linearization: A linearization is the tangent line approximation $L(x) = f(a) + f'(a)(x-a)$ used to estimate nearby function values.

Common Mistakes to Avoid

Ignoring endpoints in optimization is wrong because absolute maximum and minimum values on $[a,b]$ can occur at $x = a$ or $x = b$ , not only where $f'(x) = 0$ .
Using area when total distance is required is wrong because displacement is $\int_a^b v(t)\,dt$ , while total distance is $\int_a^b |v(t)|\,dt$ .
Differentiating related rates equations without the chain rule is wrong because variables such as $x$ and $y$ depend on time, so $\frac{d}{dt}(x^2)$ becomes $2x\frac{dx}{dt}$ .
Forgetting units is wrong because a derivative such as $\frac{dV}{dt}$ has units of volume per time, while an integral such as $\int_a^b r(t)\,dt$ has accumulated units.
Setting up area between curves as $\int_a^b (f(x) - g(x))\,dx$ without checking which function is on top is wrong because the integrand must represent a nonnegative height or use $|f(x) - g(x)|$ .

Practice Questions

1 A rectangle has perimeter $40$ meters. Write the area as a function of one variable and find the dimensions that maximize the area.
2 A particle moves with velocity $v(t) = t^2 - 4t + 3$ on $0 \le t \le 4$ . Find its displacement and total distance traveled.
3 Find the area between $y = x^2$ and $y = 2x$ on the interval where the curves intersect.
4 Explain how you would decide whether a word problem should be solved using optimization, related rates, or accumulation.