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Applications of Calculus cheat sheet - grade 11-12

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Calculus Grade 11-12

Applications of Calculus Cheat Sheet

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

TopicsDerivativesIntegralsFunctionsMotionCalculusStudy Guide
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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)=0f'(x) = 0, while related rates problems usually require differentiating an equation with respect to time tt.

Careful units, diagrams, and domain restrictions are essential in every application.

Key Facts

  • Critical numbers occur where f′(x)=0f'(x) = 0 or where f′(x)f'(x) is undefined, and candidates for absolute extrema on [a,b][a,b] include critical numbers and endpoints.
  • For optimization, write the quantity to maximize or minimize as one function, then solve f′(x)=0f'(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 ddt(x2+y2)=2xdxdt+2ydydt\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)v(t) = s'(t) and acceleration is a(t)=v′(t)=s′′(t)a(t) = v'(t) = s''(t).
  • Displacement over [a,b][a,b] is ∫abv(t) dt\int_a^b v(t)\,dt, while total distance traveled is ∫ab∣v(t)∣ dt\int_a^b |v(t)|\,dt.
  • The area between curves y=f(x)y = f(x) and y=g(x)y = g(x) on [a,b][a,b] is A=∫ab∣f(x)−g(x)∣ dxA = \int_a^b |f(x) - g(x)|\,dx.
  • The average value of a continuous function on [a,b][a,b] is favg=1b−a∫abf(x) dxf_{\text{avg}} = \frac{1}{b-a}\int_a^b f(x)\,dx.
  • A linear approximation near x=ax = a is L(x)=f(a)+f′(a)(x−a)L(x) = f(a) + f'(a)(x-a).

Vocabulary

Critical number
A critical number is a value cc in the domain of ff where f′(c)=0f'(c) = 0 or f′(c)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 ∫abr(t) dt\int_a^b r(t)\,dt.
Average value
The average value of ff on [a,b][a,b] is the constant height 1b−a∫abf(x) dx\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)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][a,b] can occur at x=ax = a or x=bx = b, not only where f′(x)=0f'(x) = 0.
  • Using area when total distance is required is wrong because displacement is ∫abv(t) dt\int_a^b v(t)\,dt, while total distance is ∫ab∣v(t)∣ dt\int_a^b |v(t)|\,dt.
  • Differentiating related rates equations without the chain rule is wrong because variables such as xx and yy depend on time, so ddt(x2)\frac{d}{dt}(x^2) becomes 2xdxdt2x\frac{dx}{dt}.
  • Forgetting units is wrong because a derivative such as dVdt\frac{dV}{dt} has units of volume per time, while an integral such as ∫abr(t) dt\int_a^b r(t)\,dt has accumulated units.
  • Setting up area between curves as ∫ab(f(x)−g(x)) dx\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)∣|f(x) - g(x)|.

Practice Questions

  1. 1 A rectangle has perimeter 4040 meters. Write the area as a function of one variable and find the dimensions that maximize the area.
  2. 2 A particle moves with velocity v(t)=t2−4t+3v(t) = t^2 - 4t + 3 on 0≤t≤40 \le t \le 4. Find its displacement and total distance traveled.
  3. 3 Find the area between y=x2y = x^2 and y=2xy = 2x on the interval where the curves intersect.
  4. 4 Explain how you would decide whether a word problem should be solved using optimization, related rates, or accumulation.

Understanding Applications of Calculus

A calculus application begins with choosing what changes and what is fixed. This is often harder than taking the derivative or integral. In a box design problem, the amount of cardboard may be fixed while the height and width can change.

In a motion problem, time is the input and position is an output. Write down the units beside every quantity. Length might be in metres, time in seconds, and volume in cubic metres.

Units act like an error check. If a rate is measured in metres per second, multiplying it by seconds should produce metres. A result with impossible units usually means the model was set up incorrectly.

Optimization is about comparing realistic possibilities, not just finding a point where a slope is zero. A business might use it to reduce material cost. An engineer might use it to choose dimensions that hold the most liquid with limited material.

A farmer might use it to plan fencing around a field. The formula must include every restriction from the situation. For example, an open box uses no lid, so its surface area expression differs from a closed box.

Values at boundaries matter because the best practical design can occur at the edge of the allowed range. Students should reject dimensions that are negative, too large for the available material, or outside the stated conditions.

Related rates describe several quantities changing at the same moment. Consider a ladder sliding down a wall. Its height decreases while its base moves away from the wall.

The ladder length stays constant, which gives the equation that connects the changing distances. The given rate is usually known only at one instant. Substitute the current measurements after differentiating the relationship, not before.

This order keeps the rate terms present. A diagram is especially useful here. Label each variable, mark the fixed quantity, and use a clear direction convention.

A negative rate has meaning. It may show that a distance, height, temperature, or amount is decreasing.

Motion graphs require careful reading because position, velocity, and acceleration tell different stories. A particle can have zero velocity while its acceleration is not zero. This happens at a turning point, where it pauses before reversing direction.

Positive velocity means motion in the chosen positive direction, not necessarily that the particle is speeding up. Speed increases when velocity and acceleration have the same sign. Graphs make this clearer.

The slope of a position graph gives velocity, while the signed area under a velocity graph gives change in position. For total distance, separate the time interval wherever velocity changes sign. Linear approximation has a similar practical role.

Near a known input, a curve can be estimated by its tangent line. It works best for small changes and smooth functions. The estimate becomes less reliable farther from the chosen point, especially near sharp bends or places where the function changes rapidly.