Archives for August 2026

Optimization: How Calculus Finds the Best Possible Outcome

August 18, 2026 by Splendid Leave a Comment

Imagine that you own a business.

You want to maximize profit.

Or perhaps you want to minimize production costs.

Maybe you’re an engineer trying to design the most efficient machine.

Or an architect attempting to use the least amount of material while maximizing available space.

These problems share a common objective:

Finding the best possible outcome.

In mathematics, this process is called optimization.

Optimization is one of the most practical applications of differential calculus because derivatives allow us to identify where a function reaches its highest or lowest values.

What is optimization?

Optimization is the process of finding the maximum or minimum value of a function.

For example:

Maximum profit.
Minimum cost.
Maximum area.
Minimum travel time.
Maximum efficiency.

The function itself describes the quantity we want to optimize.

Why are derivatives useful?

The derivative measures the rate of change.

Suppose:

$P(x)$

represents profit.

Then:

$P'(x)$

measures how quickly profit changes.

At a maximum or minimum point, something remarkable happens.

The function temporarily stops increasing or decreasing.

The slope becomes zero.

Therefore:

$P'(x)=0$

This simple observation lies at the heart of optimization.

Local maxima and local minima

A local maximum occurs when a function changes from increasing to decreasing.

A local minimum occurs when a function changes from decreasing to increasing.

At both points:

$f'(x)=0$

These points are called critical points.

Example 1: Maximizing a quadratic function

Suppose:

$f(x)=-x^2+6x+7$

Find the maximum value.

Differentiate:

$f'(x)=-2x+6$

Set the derivative equal to zero:

$-2x+6=0$

Solve:

$x=3$

Substitute into the original function:

$f(3)=-(3)^2+6(3)+7$

Therefore:

$f(3)=16$

The maximum value is 16.

Why does the slope become zero?

Imagine climbing a hill.

At the top of the hill, you stop moving upward.

For a brief instant, the ground becomes perfectly level.

That instant corresponds to:

$f'(x)=0$

Calculus identifies that exact location.

The second derivative test

How can we determine whether a critical point is a maximum or a minimum?

The second derivative provides the answer.

If:

$f''(x)<0$

the curve bends downward.

The point is a maximum.

If:

$f''(x)>0$

the curve bends upward.

The point is a minimum.

Business applications

Economists frequently optimize:

Profit.
Revenue.
Production.
Inventory.
Advertising budgets.

Suppose profit is:

$P(q)=-2q^2+100q-200$

Differentiate:

$P'(q)=-4q+100$

Setting:

$P'(q)=0$

identifies the production level that maximizes profit.

Engineering applications

Engineers optimize:

Material usage.
Structural design.
Fuel consumption.
Manufacturing efficiency.

Calculus transforms these practical problems into mathematical equations.

A general optimization strategy

Step 1: Define the function.

Step 2: Differentiate.

Step 3: Set the derivative equal to zero.

Step 4: Solve for critical points.

Step 5: Determine whether the points are maxima or minima.

The deeper philosophical idea

Optimization is about balance.

Increase too little, and opportunities are lost.

Increase too much, and inefficiencies appear.

The optimum lies between these extremes.

Calculus helps us locate that balance.

Conclusion

Optimization is one of the most important applications of calculus.

Derivatives reveal where change stops and where the best outcomes occur.

Perhaps the simplest way to remember the idea is this:

Differentiation measures change.

Optimization identifies the best place for that change to stop.

Related Rates: Understanding How Changing Quantities Influence One Another

August 17, 2026 by Splendid Leave a Comment

One of the most fascinating ideas in calculus is that many quantities in the real world do not change independently.

When one quantity changes, another quantity often changes with it.

As a balloon expands, its radius changes and its volume changes.

As a ladder slides down a wall, its height changes and its distance from the wall changes.

As a company’s production increases, inventory, costs, and revenue change simultaneously.

Calculus provides a powerful technique for analyzing these interconnected changes.

This technique is called related rates.

What are related rates?

Related rates are problems involving two or more quantities that change over time and are connected by a mathematical relationship.

Instead of asking:

What is the value of a quantity?

Related rates ask:

How quickly is one quantity changing compared with another?

In mathematical language, we study relationships between derivatives.

The fundamental idea

Suppose two variables are connected by an equation:

$x^2+y^2=25$

Both $x$ and $y$ change over time.

Therefore:

$x=x(t)$

and

$y=y(t)$

Differentiate both sides with respect to time:

$\frac{d}{dt}(x^2+y^2)=\frac{d}{dt}(25)$

Applying the chain rule:

$2x\frac{dx}{dt}+2y\frac{dy}{dt}=0$

This equation relates the rates of change of $x$ and $y$ .

Why is the chain rule essential?

Without the chain rule, related rates would not exist.

Suppose:

$y=x^2$

and $x$ changes over time.

Then:

$\frac{dy}{dt}=\frac{dy}{dx}\cdot\frac{dx}{dt}$

Applying the derivative:

$\frac{dy}{dt}=2x\frac{dx}{dt}$

Notice what happened.

The derivative with respect to $x$ became a derivative with respect to time.

The chain rule created a bridge between two changing quantities.

Example 1: An expanding circle

Suppose the radius of a circle increases at a rate of:

$\frac{dr}{dt}=3\text{ cm/min}$

The area of a circle is:

$A=\pi r^2$

Differentiate with respect to time:

$\frac{dA}{dt}=2\pi r\frac{dr}{dt}$

Suppose:

$r=5\text{ cm}$

Substitute the known values:

$\frac{dA}{dt}=2\pi(5)(3)$

Therefore:

$\boxed{\frac{dA}{dt}=30\pi\text{ cm}^2/\text{min}}$

The area increases at a rate of $30\pi$ square centimeters per minute.

Interpreting the result

Notice that the radius changes at a constant rate.

However, the area does not.

As the circle becomes larger, the same increase in radius produces larger increases in area.

This illustrates an important principle:

A constant rate in one variable does not necessarily produce a constant rate in another.

Example 2: A sliding ladder

A ladder 10 meters long leans against a wall.

The distance from the wall is:

$x$

The height on the wall is:

$y$

The relationship is:

$x^2+y^2=100$

Suppose:

$\frac{dx}{dt}=2\text{ m/s}$

and:

$x=6\text{ m}$

Find:

$\frac{dy}{dt}$

Differentiate:

$2x\frac{dx}{dt}+2y\frac{dy}{dt}=0$

First, find $y$ :

$6^2+y^2=100$

Therefore:

$y=8$

Substitute:

$2(6)(2)+2(8)\frac{dy}{dt}=0$

Simplify:

$24+16\frac{dy}{dt}=0$

Therefore:

$\boxed{\frac{dy}{dt}=-1.5\text{ m/s}}$

The negative sign indicates that the top of the ladder is moving downward.

A business example

Suppose total revenue is:

$R=P\times Q$

where:

$P$ is price.
$Q$ is quantity sold.

Suppose both price and quantity change over time.

Differentiate:

$\frac{dR}{dt}=P\frac{dQ}{dt}+Q\frac{dP}{dt}$

Revenue growth now depends on two separate rates.

Businesses frequently analyze these relationships when forecasting sales.

A manufacturing example

Suppose the volume of a cylindrical storage tank is:

$V=\pi r^2h$

If the radius and height both change over time, then:

$\frac{dV}{dt}=2\pi rh\frac{dr}{dt}+\pi r^2\frac{dh}{dt}$

Manufacturing engineers use related rates to analyze production systems, storage capacities, and material flow.

A practical problem-solving strategy

Whenever you encounter a related-rates problem:

Step 1: Identify all changing variables.

Step 2: Write the equation connecting them.

Step 3: Differentiate with respect to time.

Step 4: Substitute the known values.

Step 5: Solve for the unknown rate.

Common mistakes

Forgetting the chain rule

Incorrect:

$\frac{d}{dt}(r^2)=2r$

Correct:

$\frac{d}{dt}(r^2)=2r\frac{dr}{dt}$

Substituting values too early

Differentiate first.

Substitute numerical values later.

Otherwise, important variables may disappear.

The deeper philosophical idea

Related rates reveal something profound about the world.

Very few systems exist in isolation.

Everything is connected.

A changing radius changes an area.

A changing position changes a velocity.

A changing price changes revenue.

Calculus allows us to quantify these relationships.

Related rates versus implicit differentiation

Concept	Purpose
Implicit differentiation	Find a derivative
Related rates	Find a relationship between rates of change

Related rates often depend on implicit differentiation.

The two topics are closely connected.

Conclusion

Related rates extend the idea of derivatives beyond individual functions.

Instead of studying how a single quantity changes, they study how multiple quantities influence one another.

Perhaps the simplest way to remember the idea is this:

Ordinary derivatives describe change.

Related rates describe how one change causes another.

This insight makes related rates one of the most powerful applications of differential calculus.

Implicit Differentiation: Finding Derivatives When Functions Refuse to Behave

August 17, 2026 by Splendid Leave a Comment

Most students begin calculus by working with functions that are easy to recognize.

For example:

$y=x^2$

$y=3x^3+2x$

In these cases, the variable y is already isolated on one side of the equation.

Functions written in this form are called explicit functions.

But what happens when a relationship cannot be written so neatly?

Consider the equation:

$x^2+y^2=25$

This equation describes a circle.

Where is the function?

Which side contains y?

Which side contains x?

The variables are mixed together.

This is where implicit differentiation becomes necessary.

Explicit versus implicit functions

An explicit function isolates the dependent variable.

For example:

$y=x^3+1$

The variable y is expressed explicitly in terms of x.

An implicit function does not isolate the dependent variable.

For example:

$x^2+y^2=25$

The variables remain intertwined.

Why not simply solve for y?

We certainly can.

From:

$x^2+y^2=25$

we obtain:

$y=\pm\sqrt{25-x^2}$

However, this approach creates two separate functions.

The original equation represented the entire circle.

Separating the equation into two functions complicates the analysis.

Implicit differentiation allows us to differentiate the original equation directly.

The fundamental idea

Differentiate both sides of the equation with respect to x.

The only difference is that we must remember an important fact:

y is not a constant.

It depends on x.

Therefore, whenever we differentiate y, we must apply the chain rule.

Example 1: Differentiating a circle

Suppose:

$x^2+y^2=25$

Differentiate both sides.

The derivative of $x^2$ is:

$2x$

The derivative of y^2 requires the chain rule:

$\frac{d}{dx}(y^2)=2y\frac{dy}{dx}$

The derivative of 25 is:

$0$

Therefore:

$2x+2y\frac{dy}{dx}=0$

Solve for $\frac{dy}{dx}$ :

$2y\frac{dy}{dx}=-2x$

Therefore:

$\boxed{\frac{dy}{dx}=-\frac{x}{y}}$

What does this derivative mean?

The derivative gives the slope of the tangent line.

Notice something remarkable.

The slope depends on both x and y.

This happens because the curve is not a traditional function.

It is a geometric relationship between two variables.

Why does the chain rule appear?

Suppose:

$y=x^2+1$

Then:

$y^2=(x^2+1)^2$

The expression $y^2$ is actually a composite function.

Differentiating it requires the chain rule.

Therefore:

$\frac{d}{dx}(y^2)=2y\frac{dy}{dx}$

Implicit differentiation is impossible without the chain rule.

Example 2: An equation containing xy

Suppose:

$xy=12$

Differentiate both sides.

The left side requires the product rule:

$\frac{d}{dx}(xy)=x\frac{dy}{dx}+y$

The derivative of 12 is zero.

Therefore:

$x\frac{dy}{dx}+y=0$

Solve for the derivative:

$x\frac{dy}{dx}=-y$

Therefore:

$\boxed{\frac{dy}{dx}=-\frac{y}{x}}$

Implicit differentiation combines multiple rules

In reality, implicit differentiation is not a separate branch of calculus.

It combines previously learned techniques:

Power rule.
Product rule.
Quotient rule.
Chain rule.

All of these rules work together.

A geometric interpretation

Imagine tracing the edge of a circle.

At every point, the curve changes direction.

The derivative measures that changing direction.

The problem is that a circle cannot be represented by a single explicit function.

Implicit differentiation allows us to study the curve without rewriting the equation.

A physics example

Suppose pressure, temperature, and volume satisfy an equation such as:

$PV=nRT$

These variables are related implicitly.

If one variable changes, the others must also change.

Scientists frequently use implicit differentiation to study these relationships.

A business example

Suppose profit depends on both price and demand:

$P(Q,D)=1000$

Demand and production may change simultaneously.

Separating the variables might be impossible.

Implicit differentiation allows economists to study how one variable affects another.

A practical method

Whenever you encounter an implicit equation:

Step 1: Differentiate every term.

Step 2: Treat y as a function of x.

Step 3: Apply the chain rule whenever y appears.

Step 4: Collect all terms containing $\frac{dy}{dx}$ .

Step 5: Solve for $\frac{dy}{dx}$ .

The deeper philosophical idea

Explicit functions describe variables that willingly reveal their relationships.

Implicit equations describe variables that remain connected and refuse to separate.

Implicit differentiation allows us to study those relationships without forcing them apart.

Perhaps that is why the technique is so powerful.

It respects the relationship rather than destroying it.

Conclusion

Implicit differentiation extends the ordinary idea of derivatives to equations in which variables are intertwined.

The technique relies on familiar ideas:

Differentiate both sides.
Remember that y depends on x.
Apply the chain rule.

The result is a powerful method for analyzing circles, ellipses, physical systems, economic models, and countless other relationships.

Perhaps the simplest way to remember the idea is this:

Explicit differentiation studies functions that have already been separated.

Implicit differentiation studies relationships that remain connected.

Integration in Probability: Why Probability Distributions Use Integration

August 15, 2026 by Splendid Leave a Comment

At first glance, probability and integration seem to belong to completely different branches of mathematics.

Probability deals with uncertainty.

Integration deals with areas and accumulated quantities.

Yet modern probability theory depends heavily on integration.

Why?

Because many random events are not discrete.

They are continuous.

And whenever a quantity changes continuously, integration becomes necessary.

Discrete probability versus continuous probability

Suppose we throw a six-sided die.

Possible outcomes are:

1, 2, 3, 4, 5, and 6.

Each outcome has a probability of:

$\frac16$

We can simply add probabilities together.

No integration is needed.

Continuous probability is different

Suppose we measure a person’s height.

Possible values include:

170 cm

170.1 cm

170.15 cm

170.151 cm

There are infinitely many possibilities.

We cannot assign probabilities to individual values.

Instead, we describe probabilities using a continuous probability distribution.

Probability density functions

Continuous probability uses a probability density function.

Suppose:

$f(x)$

represents a probability density.

The probability that a value lies between:

$a$

and

$b$

is:

$\boxed{P(a\le X\le b)=\int_a^b f(x),dx}$

Integration adds all the tiny probability contributions between two boundaries.

Why can’t we assign a probability to a single point?

In continuous distributions:

$P(X=x)=0$

This seems strange.

How can every individual probability equal zero?

Because probabilities are accumulated over intervals rather than individual points.

The probability of selecting exactly 170.000000 cm is essentially zero.

The probability of selecting a height between 170 cm and 180 cm is meaningful.

A geometric interpretation

Think of a probability distribution as a curve.

The total area under the curve must equal:

$1$

Why?

Because the probability of all possible outcomes must equal 100%.

Therefore:

$\boxed{\int_{-\infty}^{\infty}f(x),dx=1}$

The entire probability distribution is simply an accumulated area.

The normal distribution

One of the most famous probability distributions is the normal distribution.

The bell-shaped curve describes many natural phenomena:

Heights.
Examination scores.
Measurement errors.
Manufacturing variations.

The probability of observing a value within an interval equals the area under the curve.

Expected value

Probability uses integration to calculate averages.

The expected value of a continuous random variable is:

$E(X)=\int_{-\infty}^{\infty}xf(x),dx$

This formula looks remarkably similar to the average-value formula from calculus.

That is not a coincidence.

Expected value is simply a weighted average.

A business example

Suppose a company estimates customer spending using a probability distribution.

The expected customer expenditure becomes:

$E(X)=\int xf(x),dx$

Integration allows businesses to estimate:

Average customer spending.
Expected profit.
Financial risk.

A manufacturing example

Suppose the diameters of machine components follow a normal distribution.

Integration allows engineers to estimate the probability that a component falls within acceptable tolerances.

This helps improve quality control.

Why integration appears everywhere in probability

Integration performs accumulation.

Probability distributions require accumulation.

The connection is unavoidable.

Every small probability density contributes to a larger probability.

Integration combines those contributions.

The deeper philosophical idea

Probability asks:

What could happen?

Integration asks:

How much accumulated?

Continuous probability combines these questions.

It asks:

How much probability accumulated within a particular interval?

Conclusion

Probability distributions use integration because probabilities in continuous systems cannot be calculated by simple addition.

Instead, probabilities accumulate continuously.

Integration provides the mathematical framework needed to measure that accumulation.

Perhaps the simplest way to remember the relationship is this:

Probability densities describe how probability is distributed.

Integration measures how much probability accumulates.

And that is why integration lies at the heart of modern probability theory.

Integration in Economics: Understanding Consumer Surplus, Producer Surplus, Total Revenue, and Total Cost

August 15, 2026 by Splendid Leave a Comment

Many students learn integration by calculating areas under curves and solving mathematical exercises.

Economists, however, use integration for a very different purpose.

They use it to answer questions such as:

How much value do consumers receive from a market?
How much benefit do producers obtain from selling their products?
How can we reconstruct total revenue from marginal revenue?
How can we reconstruct total cost from marginal cost?

In economics, integration transforms rates, prices, and marginal quantities into meaningful business information.

The reason is simple.

Most economic variables change continuously.

Calculus provides the language needed to analyze those continuous changes.

Why economics needs integration

Suppose a company sells products.

The first product might cost very little to manufacture.

The hundredth product might cost much more.

Costs change continuously.

Revenue changes continuously.

Demand changes continuously.

Simple arithmetic is often insufficient.

Integration allows economists to accumulate these changing quantities over time or over production levels.

Marginal quantities and total quantities

Economics frequently uses the word marginal.

Marginal means:

The change resulting from one additional unit.

For example:

Marginal cost:

$MC(q)=\frac{dC}{dq}$

Marginal revenue:

$MR(q)=\frac{dR}{dq}$

Differentiation converts total quantities into marginal quantities.

Integration reverses the process.

Reconstructing total cost from marginal cost

Suppose marginal cost is:

$MC(q)=50+4q$

This means the cost of producing one additional unit increases as production increases.

Because:

$MC(q)=\frac{dC}{dq}$

we can recover total cost by integrating:

$C(q)=\int(50+4q),dq$

Therefore:

$C(q)=50q+2q^2+C$

Integration reconstructs the total cost function.

Reconstructing total revenue from marginal revenue

Suppose:

$MR(q)=100-2q$

Total revenue is:

$R(q)=\int(100-2q),dq$

Integrating:

$R(q)=100q-q^2+C$

The revenue function has been reconstructed from its rate of change.

Consumer surplus

Consumer surplus is one of the most important applications of integration in economics.

Consumers are often willing to pay more than the market price.

The difference between what consumers are willing to pay and what they actually pay represents consumer surplus.

A demand curve example

Suppose the demand function is:

$P_d=100-Q$

The market price is:

$P=40$

The quantity demanded is:

$Q=60$

Consumer surplus equals the area between the demand curve and the market price.

Therefore:

$CS=\int_0^{60}[(100-Q)-40],dQ$

Simplifying:

$=\int_0^{60}(60-Q),dQ$

Integrating:

$=\left(60Q-\frac{Q^2}{2}\right)_0^{60}$

Therefore:

$=1800$

Interpreting consumer surplus

The number 1800 does not represent sales revenue.

It represents additional value received by consumers.

In other words:

Consumers collectively gained 1800 monetary units beyond what they paid.

Producer surplus

Producer surplus measures the benefit producers receive.

Suppose the supply function is:

$P_s=20+\frac{Q}{3}$

The market price remains:

$P=40$

Producer surplus is the area between the market price and the supply curve.

Therefore:

$PS=\int_0^{60}\left[40-\left(20+\frac{Q}{3}\right)\right],dQ$

Integrating gives the producer surplus.

Why economists love integration

Integration converts pictures into numbers.

A graph showing demand and supply becomes a measurable economic quantity.

Areas become:

Consumer surplus.
Producer surplus.
Total revenue.
Total cost.
Total welfare.

A business interpretation

Imagine running an online business.

Each additional customer generates different costs.

Each additional sale generates different revenue.

Integration allows businesses to answer questions such as:

What are our total costs?
What is our accumulated revenue?
How much value are customers receiving?

Calculus becomes a practical decision-making tool.

The deeper connection

Differentiation asks:

How rapidly are costs changing?

Integration asks:

How much cost has accumulated?

Differentiation asks:

How rapidly is revenue changing?

Integration asks:

How much revenue has accumulated?

Economics constantly moves between these two perspectives.

Conclusion

Integration plays a central role in economics.

It reconstructs total quantities from marginal quantities.

It measures consumer and producer benefits.

It converts changing economic relationships into meaningful business information.

Perhaps the simplest way to summarize the idea is this:

Differentiation measures economic change.

Integration measures accumulated economic value.

Average Value of a Function: How Integration Finds the Typical Behavior of a Changing Quantity

August 15, 2026 by Splendid Leave a Comment

When someone asks for an average, we usually think of arithmetic.

Suppose a student scores:

80, 90, and 100.

The average is:

$\frac{80+90+100}{3}=90$

But what happens when the quantity changes continuously?

How do we calculate the average temperature during a day?

The average stock price during a month?

The average speed of a moving vehicle?

This is where integration becomes essential.

Revisiting the ordinary average

Suppose we have the numbers:

4, 6, and 8.

The average is:

$\frac{4+6+8}{3}=6$

The arithmetic mean distributes the total equally.

You can visualize this idea here:

Integration extends this idea from discrete data to continuously changing functions.

The average value of a continuous function

Suppose:

$y=f(x)$

between:

$x=a$

and

$x=b$

The total accumulated value is:

$\int_a^b f(x),dx$

However, an average requires equal distribution.

Therefore, we divide by the interval length:

$\boxed{f_{\text{avg}}=\frac{1}{b-a}\int_a^b f(x),dx}$

Why divide by b-a?

Integration gives the total accumulation.

An average requires us to spread that accumulation evenly.

Dividing by the interval length accomplishes exactly that.

Example 1: Finding an average value

Suppose:

$f(x)=x^2$

between:

$0\le x\le3$

First, calculate the integral:

$\int_0^3x^2,dx$

Integrating:

$=\left(\frac{x^3}{3}\right)_0^3$

Evaluating:

$=9$

Now divide by the interval length:

$f_{\text{avg}}=\frac{9}{3}$

Therefore:

$f_{\text{avg}}=3$

Average speed

Suppose velocity is:

$v(t)=2t$

between:

$t=0$

and

$t=5$

The average velocity is:

$\frac{1}{5}\int_0^52t,dt$

The integral is:

$25$

Therefore:

$=\frac{25}{5}$

The average velocity is:

$5$

Average cost in economics

Suppose total cost is:

$C(q)=100+5q+q^2$

A manufacturer wants to know the average cost over a production interval.

Integration provides a more realistic measurement because costs often change continuously.

Average stock prices

Stock prices fluctuate constantly.

Instead of examining a few isolated observations, analysts often model prices as continuous functions.

Integration can estimate the average value over a trading period.

Average temperature

Meteorologists rarely rely on a few measurements.

Temperature changes continuously.

Integration allows them to calculate the average temperature during an entire day.

Average values in probability

Probability theory frequently uses average values.

The expected value of a continuous random variable is based on integration.

In fact, expected value is simply another type of weighted average.

The deeper connection

An ordinary average works with individual data points.

An integral works with continuous accumulation.

The average value formula combines both ideas.

It first accumulates everything.

Then it redistributes the total equally.

The key insight

The average value of a function can be understood in three steps:

Accumulate everything.
Measure the interval.
Distribute the accumulation equally.

Conclusion

Integration doesn’t merely calculate areas.

It helps us describe the typical behavior of continuously changing quantities.

Whether we’re measuring speed, temperature, stock prices, production costs, or probabilities, the underlying idea remains the same:

Accumulate first.

Average second.

This elegant connection between integration and averaging reveals yet another practical application of calculus.

Archives for August 2026

What is optimization?

Why are derivatives useful?

Local maxima and local minima

Example 1: Maximizing a quadratic function

Why does the slope become zero?

The second derivative test

Business applications

Engineering applications

A general optimization strategy

The deeper philosophical idea

Conclusion

What are related rates?

The fundamental idea

Why is the chain rule essential?

Example 1: An expanding circle

Interpreting the result

Example 2: A sliding ladder

A business example

A manufacturing example

A practical problem-solving strategy

Common mistakes

Forgetting the chain rule

Substituting values too early

The deeper philosophical idea

Related rates versus implicit differentiation

Conclusion

Explicit versus implicit functions

Why not simply solve for y?

The fundamental idea

Example 1: Differentiating a circle

What does this derivative mean?

Why does the chain rule appear?

Example 2: An equation containing xy

Implicit differentiation combines multiple rules

A geometric interpretation

A physics example

A business example

A practical method

The deeper philosophical idea

Conclusion

Discrete probability versus continuous probability

Continuous probability is different

Probability density functions

Why can’t we assign a probability to a single point?

A geometric interpretation

The normal distribution

Expected value

A business example

A manufacturing example

Why integration appears everywhere in probability

The deeper philosophical idea

Conclusion

Why economics needs integration

Marginal quantities and total quantities

Reconstructing total cost from marginal cost

Reconstructing total revenue from marginal revenue

Consumer surplus

A demand curve example

Interpreting consumer surplus

Producer surplus

Why economists love integration

A business interpretation

The deeper connection

Conclusion

Revisiting the ordinary average

The average value of a continuous function

Why divide by b-a?

Example 1: Finding an average value

Average speed

Average cost in economics

Average stock prices

Average temperature

Average values in probability

The deeper connection

The key insight

Conclusion

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Calculus 1A: Differentiation

Calculus 1B: Integration