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FFR - Applications

This original textbook reading covers part of the AP topic. Follow the AP course outline for complete topic instruction and practice.

This reading includes verified answer corrections and explicitly stated clarifications.

How do we use derivatives in real life?

There are many natural processes that are governed by what we have been studying most in higher level in mathematics: functions. The decay of uranium, the increasing balance due to compounded interest in your bank account, the increasing amount you owe the bank due to mortgage, the decrease in your purchasing power due to inflation, the rotation of the earth around the sun in an elliptical orbit, among many others. We are going to decode how to translate these processes from math to english and vice versa. Here are some key words that indicate that you must take the derivative of a function.

Buzz words for a derivative:

•Change in some variable (ex: pressure, concentration, velocity) with respect to another variable (ex: altitude, time, distance)

•Instantaneous

•Rate of increase/decrease

•Tangential

As we get farther, we will learn to visualize what a question is asking from more than just the above key words. Let’s learn by doing though.

Example Problem 5.4.1

Let’s use an idealized carbon-14 decay model with a half-life of about 5700 years. The independently supplied initial amount is B = 4 g, and the present amount is M = 10 mg = 0.01 g. Assume no isotope enters or leaves the sample after the model begins, apart from radioactive decay. These supplied model amounts replace the earlier unsupported calculation from a mammoth’s total body mass. Real radiocarbon dating uses the remaining isotope ratio in once-living material and calibration; we cannot infer an initial carbon-14 amount from total fossil mass alone.

a) According to this decay model, how much time has elapsed?

b) What is the current rate of change of the tracked carbon-14 mass, in mg per millennium?

a) This seems like a really long question, but we can break this down piece by piece. First of all, let’s define half-life. It’s the time it takes for half the mass (or concentration in chemistry) to be consumed. Half life stays constant for first order exponential decay.

The initial amount B = 4 g is a supplied model parameter. It does not follow from the total body mass of an animal. For a dating model, the initial isotope amount or ratio must be established independently before the remaining fraction can determine elapsed time.

After 1 half-life cycle, this mass will become 2g, then after another 5700 years, it’ll become 1g, then 0.5g after another 5700 years, and so on. That’s what the formula M=Be^-rt represents. B represents initial mass when the decay started to occur. M represents present mass. t represents the amount of time that has passed, and r is the rate constant.

As the name implies, r stays constant, but we do not know what it is. How can we figure it out?

We know the definition of a half life, so let’s convert that into equation form:

$1 = 2 e^{- r \cdot 5.7}$ (I used 5.7 here because keeping the units in millennia makes it easier to calculate than keeping them in years. r will just have different units, but it will still get the job done).

Since B represents the initial mass, we just gave it an arbitrary number (2 is the easiest) so that the final mass will be half of B (in this case 1). Now let’s do some algebraic manipulation that gives us
$\ln \frac{1}{2} = \ln e^{- r \cdot 5.7}$
$- 0.693 = - r \cdot 5.7$
$r = 0.1216 {millenia}^{- 1}$

It’s important to keep track of units in problems like these. Had we chosen to go with years instead of millennia, r would be a much smaller value (1000 times smaller) with different units (years^-1). A general rule of thumb is that the exponent should have no units (units should cancel out).

Now use consistent units: B = 4 g, M = 10 mg = 0.01 g, and t is in millennia. The half-life gives r = ln(2)/5.7 ≈ 0.1216 per millennium. Substitute those values into M = B e^−rt.

$0.01 = 4 \cdot e^{- 0.1216 \cdot t}$

$\ln (0.0025) = \ln (e^{- 0.1216 \cdot t})$

Taking logarithms gives ln(0.0025) = −rt, so t = ln(400)/r. Keep the unrounded logarithm until the final calculation.

Using r = ln(2)/5.7, t = 5.7 ln(400)/ln(2) ≈ 49.2700 millennia, or about 49,270 years. With the displayed rounded constant 0.1216, the answer is about 49.2719 millennia. Both round to 49.3 millennia. This is the idealized model’s elapsed time, not a calibrated laboratory date.

b) To find the current rate of decay, we must take the derivative with respect to time, then plug in the current time relative to when the decay started. First let’s take the derivative of the function.

$M = 4 e^{- 0.1216 \cdot t}$
$\frac{d M}{d t} = - 0.1216 \cdot 4 e^{- 0.1216 \cdot t}$ (chain rule)

At the current mass M = 0.01 g, use M′ = −rM. With the displayed r ≈ 0.1216 per millennium, M′ ≈ −0.1216 · 0.01 = −0.001216 g per millennium = −1.216 mg per millennium. The negative sign means the mass is decreasing; the positive magnitude is the decay rate.

The negative simply means that the mass is decreasing.

Interactive 5.4.1 Sample Problem 5.4.1

Magnification 1×

Read all original figure notes

Model positivity and measurement limits

In the idealized exponential model, M(t) stays positive at every finite time and approaches 0 as t grows. That does not mean an arbitrarily old sample has measurable carbon-14. Real radiocarbon dating needs a remaining isotope ratio, an initial-ratio model, and calibration, and is used for once-living material only up to about 50,000 years—not the age of the non-avian dinosaurs. Our supplied initial and present amounts define a mathematics model, not a laboratory date. See the National Park Service explanation of radiocarbon dating.

Half-Life 1

On the y-axis we have the ratio of final to initial mass. Notice that when 50% of the mass has depleted, the x-value (time) is approximately equal to 5.7 millennia

Half-Life 2

Again notice that the amount of time to deplete half the previous amount has stayed the same.

Show your work for Problem 1

You deposit $1000 into the stock market that you plan to keep until you retire in 45 years. Your portfolio has an annual rate of return of 10% per year. If you are 20 years old, at what rate is your wealth growing when you are 60? (Use units of dollars per year)

Clarification: Assume continuous compounding at an annual rate of 0.10, with t in years since the deposit.

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Show your work for Problem 2

Find

lim_{x \to 3} \frac{x - 3}{\sqrt{x - 3}} + x \cdot \ln (x - 3) - 3 \ln (x - 3)

Clarification: Take the real-domain limit as x approaches 3 from the right.

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Chapter 5 Problems

You deposit $1000 into the stock market that you plan to keep until you retire in 45 years. Your portfolio has an annual rate of return of 10% per year. If you are 20 years old, at what rate is your wealth growing when you are 60? (Use units of dollars per year)
Clarification: Assume continuous compounding at an annual rate of 0.10, with t in years since the deposit.
Find $lim_{x \to 3} \frac{x - 3}{\sqrt{x - 3}} + x \cdot \ln (x - 3) - 3 \ln (x - 3)$
Clarification: Take the real-domain limit as x approaches 3 from the right.

Show your work for Problem 4

What is

lim_{x \to \infty} \sin (\frac{\ln x}{x})

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Show your work for Problem 3

Solve

lim_{x \to \infty} x^{- x^{x}}

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Solve $lim_{x \to \infty} x^{- x^{x}}$
What is $lim_{x \to \infty} \sin (\frac{\ln x}{x})$

Show your work for Problem 6

Find the derivative of

f (x) = \tan^{- 1} (\log_{2} x) + 3 \sin^{- 1} (\cos^{- 1} x)

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Show your work for Problem 5

Calculate the derivative of

f (x) = \log_{4} (x^{2}) + 3^{x \cdot \log_{2} (x)} + 4 \ln (x)

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Calculate the derivative of $f (x) = \log_{4} (x^{2}) + 3^{x \cdot \log_{2} (x)} + 4 \ln (x)$
Find the derivative of $f (x) = \tan^{- 1} (\log_{2} x) + 3 \sin^{- 1} (\cos^{- 1} x)$

Show your work for Problem 8

Find the derivative of

f (x) = 5 \sec^{- 1} (4^{x}) + 2 \cot^{- 1} (\csc^{- 1} (x)) + 8^{x^{\ln x}}

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Show your work for Problem 7

Find

lim_{x \to \infty} {(4 + \frac{3}{x})}^{\frac{1}{x}}

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Find $lim_{x \to \infty} {(4 + \frac{3}{x})}^{\frac{1}{x}}$
Find the derivative of $f (x) = 5 \sec^{- 1} (4^{x}) + 2 \cot^{- 1} (\csc^{- 1} (x)) + 8^{x^{\ln x}}$

Show your work for Problem 10

Find the slope of the tangent line at x=0.5 of

f (x) = 3 \sec^{- 1} (4^{\log_{2} x}) + 2 \cos^{- 1} (\sin^{- 1} (x)) + 2^{x \cdot \ln (3 x)}

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Show your work for Problem 9

Find the slope of the tangent line at x=0 of

f (x) = 2 \sin^{- 1} (x) + 3^{x}

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Find the slope of the tangent line at x=0 of $f (x) = 2 \sin^{- 1} (x) + 3^{x}$
Find the slope of the tangent line at x=0.5 of $f (x) = 3 \sec^{- 1} (4^{\log_{2} x}) + 2 \cos^{- 1} (\sin^{- 1} (x)) + 2^{x \cdot \ln (3 x)}$

Glossary and deeper reading · 3 refreshers

These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.

Function
A mathematical function assigns exactly one output to each input in its domain. Inputs can share an output, but one input cannot have two different outputs. For example, f(x)=x² assigns the output 9 to both 3 and −3. Function notation names the rule and its input; f(3) means the output when the input is 3.
Derivative
A derivative gives the instantaneous rate of change of a function with respect to its input. At a point, it is the finite limit of the difference quotient when that limit exists. For example, the derivative of x² is 2x, so the tangent slope at x=3 is 6. A derivative function collects those rates across the inputs where the function is differentiable.
Read more: 2.2 · Derivative →
Velocity in calculus
In one-dimensional motion, velocity is the derivative of position with respect to time. Its sign records direction relative to the chosen positive axis. For example, s(t)=t²−4t meters gives velocity 2t−4 meters per second. Speed is the magnitude of velocity and therefore cannot be negative.
Read more: 4.2 · Velocity in calculus →

Interactive 5.4.1 Sample Problem 5.4.1

Model positivity and measurement limits

Half-Life 1

Half-Life 2

Show your work for Problem 1

Show your work for Problem 2

Show your work for Problem 4

Show your work for Problem 3

Show your work for Problem 6

Show your work for Problem 5

Show your work for Problem 8

Show your work for Problem 7

Show your work for Problem 10

Show your work for Problem 9

Function

Derivative

Velocity in calculus

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