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B. Derivatives and Their Applications · Applications of Derivatives: Motion, Rates and Optimization

Applications of Derivatives: Motion, Rates and Optimization: Unit Review

B2.1 B2.2 B2.3 B2.4 B2.5About 40 minutes
Learning goals
  • I can solve motion, rate-of-change and related-rates problems with derivatives.
  • I can solve optimization problems with polynomial, simple rational and exponential models.
  • I can build, solve, check and communicate a mathematical model.

Words to know

velocity
v(t) = s′(t).
acceleration
a(t) = v′(t) = s″(t).
marginal
In business, the derivative: the approximate effect of one more item.
optimization
Finding the absolute maximum or minimum of a quantity on its domain.

Let's review!

In this unit you used derivatives to describe and improve the real world. You found velocity and acceleration as the first and second derivatives of position, decided when objects speed up or slow down, and read motion graphs. You interpreted derivatives as real-world rates (inflation, population, flow), found instantaneous rates in growth, decay and tide models, used marginal cost, revenue and profit, and linked rates with the chain rule in related rates. Then you optimized: absolute extremes, areas and volumes, least cost and material, best prices, fastest routes, and finally the full modelling cycle with the foraging bird.

This review has a few questions from every lesson. Try each one, then press Show answer. The next unit, Vectors in Two-Space and Three-Space, starts the third strand of the course.

Motions, v = s′, a = s″ (1 to 3)Rates in contextunits, inflation (4 to 5)Marginal, relatedbusiness, chain rule (6, 7)Absolute max and mincritical numbers + endpoints (8)Optimization problemsgeometry, cost, business, time (9 to 12)Modellingsolve, interpret, validate, communicate (13)
How the unit fits together. The motion and rate lessons interpret derivatives; the absolute-extremes lesson gives the tool; the optimization and modelling lessons use it.

How to use this review

Work one or two sections at a time with a scientific calculator in radian mode. For each problem, first say what is being asked: a rate at an instant, a time when something happens, or a best value. Draw a diagram for every related-rates and optimization problem, write the domain, and finish every answer with a sentence and units. For each mistake, write down why it happened (wrong derivative? forgot an endpoint? units?) and revisit that lesson. Finish with the 10-question quiz.

What does the problem ask?(read it twice)"how fast?" at an instantdifferentiate, substitute,answer with units"when is it at rest / fastest?"solve v = 0 or a = 0,sign chart"largest / least / best?"one-variable function,domain, test candidates
A quick way to choose a method: an instantaneous rate needs a derivative evaluated at one point; "when" questions need you to solve an equation; "best" questions are optimization.

Position, Velocity and Acceleration as Derivatives

1

s(t) = t³ − 9t² + 24t (m, s). Find v(t) and a(t).

2

For the same s(t), when is the object at rest?

3

A ball's height is h(t) = −4.9t² + 24.5t + 1 (m). Find its maximum height.

4

s(t) = 3 sin(2t) cm. Find a(t) and compare it with s(t).

Speeding Up, Slowing Down, and Total Distance

1

v(3) = −2 and a(3) = 5. Speeding up or slowing down?

2

s(t) = t² − 6t (m, s). Find the total distance travelled from t = 0 to t = 5.

3

For s(t) = t³ − 3t² (t ≥ 0), find when the object is speeding up.

4

An object travels 30 m forward and 10 m back in 8 s. Find its average velocity and average speed.

Motion Graphs: From Position to Velocity and Acceleration

1

An s-t graph is concave down and rising. Give the signs of v and a.

2

A v-t graph crosses the t-axis from below to above at t = 4. What does the object do there?

3

s(t) = −t³ + 9t². When is the velocity greatest?

4

A walker's s-t graph is a straight line from (0, 5) to (5, 1). Find v.

Derivatives as Real-World Rates: Inflation, Population and Flow

1

W(t) is the number of litres of water in a rain barrel t hours after a storm starts. Interpret W′(3) = 12.

2

A price index is P(t) = 100 + 3t² − 0.2t³ (0 ≤ t ≤ 15). When is inflation fastest?

3

A city of 250 000 grows by 5000 people per year. Find the relative growth rate.

4

If f′(t) < 0 and f″(t) > 0, describe f.

Instantaneous Rates in Models: Growth, Decay and Tides

1

P(t) = 400e^(0.03t). Find P′(10).

2

A(t) = 50(0.5)^(t/6) mg. Find A′(6).

3

d(t) = 4 + 1.5 cos(πt/6) m (t in hours). Find the greatest rate of change of depth.

4

C(t) = 8t/(t² + 9). Find C′(1).

Marginal Cost, Revenue and Profit

1

C(x) = 800 + 6x + 0.02x². Find the marginal cost at x = 100.

2

For the same C(x), find the average cost at x = 100.

3

p(x) = 40 − 0.05x. Find R′(x) and R′(300).

4

Marginal cost is $14 and average cost is $18. Is the average cost rising or falling?

Related Rates: Linking Rates with the Chain Rule

1

A square's side grows at 3 cm/s. How fast is its area growing when the side is 7 cm?

2

A spherical balloon's radius grows at 0.5 cm/s. Find dV/dt when r = 6 cm.

3

A 13 m ladder slides; its bottom moves away at 1 m/s. How fast does the top fall when the bottom is 5 m from the wall?

4

Water enters a cylinder of radius 2 m at 0.8 m³/min. How fast does the level rise?

Optimization Basics: Absolute Maximum and Minimum Values

1

Find the absolute extremes of f(x) = x³ − 6x² + 5 on −1 ≤ x ≤ 5.

2

Find the absolute minimum of g(x) = x + 36/x on 1 ≤ x ≤ 10.

3

Find the maximum of h(t) = 8te^(−t/3) for t ≥ 0.

4

Why must endpoints be checked on a closed interval?

Optimization in Geometry: Areas and Volumes

1

With 240 m of fence, enclose a rectangle along a barn wall (three sides fenced). Find the largest area.

2

An open box is made from an 18 cm by 18 cm sheet by cutting x-by-x corners. Find the best x.

3

A rectangle has its base on the x-axis and upper corners on y = 48 − 3x². Find its largest area.

Optimization: Minimizing Cost and Materials

1

A closed cylinder holds 2000 cm³. Find the radius for least surface area.

2

C(v) = 16 000/v + 10v for 30 ≤ v ≤ 100. Find the cheapest speed.

3

A square-based open box holds 500 cm³. Find the dimensions of least material.

Optimization in Business: Maximizing Revenue and Profit

1

A gym has 500 members at $40 a month. Each $2 increase loses 10 members. Find the price that maximizes revenue.

2

p = 90 − 0.3x and C = 600 + 30x. Find the quantity that maximizes profit.

3

Demand p = 25e^(−0.02x). Find the revenue-maximizing quantity.

Optimization: Time, Distance and Exponential Models

1

A swimmer is 1 km from shore; the finish is 4 km along the shore. She swims at 3 km/h and runs at 5 km/h. Where should she land?

2

Two cyclists leave the same corner: one goes north at 12 km/h, the other east at 16 km/h. Is there a time of closest approach after they start?

3

Find the point on y = x² closest to (0, 2).

Modelling with Derivatives: The Foraging Bird and Other Real Problems

1

For E(t) = 40t/(t + 4) and travel time T = 9, find the best stay.

2

Explain the marginal value rule in one sentence.

3

Name two things a good written conclusion to a modelling task must include besides the answer.

4

Why is it risky to use a model far outside the data range?

Check your understanding

Quiz

1.

s(t) = 2t³ − 3t² (m, s). What is a(2)?

2.

At an instant, v = 4 m/s and a = −3 m/s². The object is…

3.

s(t) = t² − 4t (m) for 0 ≤ t ≤ 6. What is the total distance travelled?

4.

A price graph is rising and its point of inflection is at t = 3. What happens at t = 3?

5.

A(t) = 80e^(−0.1t). What is A′(t)?

6.

The radius of a circle grows at 2 m/s. How fast is the area growing when r = 10 m?

7.

For a continuous f on [a, b], the absolute maximum must occur…

8.

A closed cylinder of fixed volume has least surface area when…

9.

Demand is p = 30 − 0.1x. Which quantity maximizes revenue?

10.

In the foraging model R(t) = E(t)/(t + T), what happens to the best stay when T increases?

For parents and teachers

How to use this review at home: have your teen do one section at a time and mark it. Each section matches one lesson, so a wrong answer tells you exactly where to go back.

Which lesson to revisit:

  • Confusing position with velocity, or units of acceleration: Position, Velocity and Acceleration as Derivatives.
  • Wrong "speeding up / slowing down" verdicts or distance errors: Speeding Up, Slowing Down, and Total Distance.
  • Misreading slopes on motion graphs: Motion Graphs.
  • Missing units or mixing a quantity with its rate: Derivatives as Real-World Rates.
  • Chain-rule slips with eˣ, aˣ, sin and cos models, or degree mode: Instantaneous Rates in Models.
  • Mixing up revenue, price, average cost and marginal cost: Marginal Cost, Revenue and Profit.
  • Substituting too early in related rates: Related Rates.
  • Forgetting endpoints or out-of-domain critical numbers: Optimization Basics.
  • Set-up errors (two variables left, wrong constraint): Optimization in Geometry and Minimizing Cost and Materials.
  • Price-increase set-ups and profit versus revenue: Optimization in Business.
  • Time = distance ÷ speed, distance formula: Optimization: Time, Distance and Exponential Models.
  • Answers with no interpretation, assumptions or limits: Modelling with Derivatives.