### Calculus in a Nutshell

Calculus means “small pebble,” as in the primary part of an abacus, a calculator from the old days.

However, while an abacus is pretty much limited to arithmetic that's ticked out discretely with the click of stones, calculus was invented to study the smoothly varying behavior of moving objects.

In the time since its inception, calculus has grown into a tool used to study all kinds of change. Many scientists, engineers, and analysts rely heavily on it to model and understand the systems they study.

This quiz introduces the first of three important pillars of calculus: the derivative.

# Derivatives in a Nutshell

Before we define derivatives, let's quickly look at what they're used for.

Derivatives help solve many optimization problems in which the output of a given function needs to be maximized or minimized.

Here's a problem we'll actually crack with derivatives later in the course:

A soccer player makes a breakaway, running straight up the field. Her two lines of sight to each of the goal posts form an angle $\theta$ that changes with time $t$ as she runs.

The interactive plot below shows the setup. Press play and watch $\theta$ change with time, or manually adjust the slider to any desired $t.$

The player has her best chance of scoring when the angle $\theta$ is largest. Is there an optimal time for her to shoot?

# Derivatives in a Nutshell

Optimization is one of the most important applications of calculus in many fields of science and engineering.

To start us on the path to optimization with calculus, let's look at an example where you can eyeball the maximum from the function's graph.

A cannonball is fired directly into the air. Its height above the ground is a function of time, $h(t).$

Use the plot to estimate when it reaches its maximum height.

# Derivatives in a Nutshell

The graph below shows the same cannonball trajectory, but now it displays the parabola together with the line just grazing it at $\big(t,h(t)\big).$

This line is called the tangent line, and its position and slope change as you change $t$ in the plot below.

What is the tangent line's slope when the cannonball reaches its max height?

# Derivatives in a Nutshell

The tangent line of $h(t)$ at the point $t=a$ is the one line that just grazes $h$'s graph at the point $\big (a, h(a) \big)$ without slicing through it.

For any smooth curve, there is only one such line for each point of the curve since, if you zoom in closer and closer to any point, you can see the curve has a "local slope" there.

Only a tangent line with exactly that same slope can touch the curve there without piercing it.

# Derivatives in a Nutshell

A straight line can only be tangent to a curve if its slope exactly matches how quickly the function itself is changing at the point of tangency.

Fill in the blank:

Any line that is tangent to a peak of a smooth function's graph will be perfectly $\text{\_\_\_\_\_}.$

# Derivatives in a Nutshell

Because it can do so much for us, the slope of a tangent line is given a special name: "derivative." And "taking the derivative" of a function at a point means finding the slope of the tangent line there.

For example, the slope of the tangent line of $h$ at the point $\big (t, h(t) \big)$ (see graph below) starts at $10$ when $t=0$, but steadily decreases as you move up and over the curve's peak.

If you move the slider to any value of $t$ in the graph below, you can see that $v(t)=10-10t$ precisely equals the slope of the tangent line.

So we would say that the derivative of $h$ at $\big (t,h(t) \big)$ is $v(t) = 10 - 10 t.$

What is the derivative of $h$ when $t = \frac{1}{2} \ ?$

# Derivatives in a Nutshell

In a nutshell: Derivatives measure rates of change.

They play a central role in kinematics, the study of motion: the derivative of a position function like $h(t)$ is a velocity function that describes the varying velocity of the object. In fact, that's why we chose to name function for tangent line's slope $v(t)$ in the last problem!

This and the fact that derivatives solve optimization problems show that the skill of calculating derivatives is one of the most important takeaways from any course on calculus.

The name may seem strange, but we'll come to understand where the term comes from as the course develops.

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