Topic 5.8: Sketching Graphs of Functions and Their Derivatives

The twinkle

This is the top of Cerro Paranal, a mountain in the Atacama Desert in Chile, and those four buildings hold some of the biggest telescopes on Earth, each one with a mirror more than eight meters across. Those orange beams are lasers, and each one lights up a thin layer of sodium atoms about ninety kilometers up, which gives the telescope its own artificial stars to watch, bright points of light its sensor can always see.

Why would anyone need a fake star? Every star you’ve ever seen twinkles, and it isn’t the star doing it. Its light travels for years without a scratch, and then the last twenty kilometers of our own air smear it. Picture the starlight arriving as flat sheets, and the moving air putting dents in them. The telescope can’t measure those dents directly, though, because its sensor only measures slopes, how tilted the light is under each one of its tiny lenses.

From nothing but those slopes, a computer works out the shape of the dent and bends a mirror just two millimeters thick to cancel it, about a thousand times every second. Look what that does to Neptune. By the end of this lesson, you’ll be able to do what that computer does. You’ll look at nothing but a graph of slopes and draw the shape it came from, every peak, every valley and every place it bends.

Slopes become heights

Let’s freeze the light for a single instant and take one slice through it. The blue curve is the height of the light across the beam, and under it sits a row of tiny lenses, each one focusing its little piece of light down to a bright dot.

Where the light is tilted steeply, like right here at the left edge, the dot slides a long way, and light tilted the other way slides its dot the other way. Up at the crest, where the light is flat right there, the dot sits dead center. Now let’s take every dot’s shift and stand it up as a height, and the heights line up into a brand new curve, which is the only thing the computer ever gets to see.

That new curve is a graph you already know, the derivative of the blue curve, f prime, so let’s take both of them to the board and read them carefully. Our slice of light is f of x equals x cubed over three, minus two x squared, plus three x. The power rule gives f prime of x equals x squared minus four x plus three, which factors into x minus one, times x minus three.

At x equals zero the tangent line on f climbs with a slope of three, and right below it the orange graph sits at a height of exactly three. Slide along and it keeps happening. At one the tangent is flat and f prime is zero, and at two the curve falls with a slope of negative one, so f prime is down at negative one.

At three it’s flat again, so f prime is back to zero, and at four the slope is three, just like at the start. If you remember one thing from today, make it this, the height of the bottom graph is the slope of the top graph, at every single x.

Let me ask you something before we go on. The orange graph, f prime, reaches its lowest point at x equals two, so what do you think the blue curve f is doing right at that spot?

We can do the exact same move one level down. The slope of the orange graph becomes the height of a third graph, the second derivative, f double prime. Since f prime is x squared minus four x plus three, f double prime is two x minus four. f double prime tells us how the slopes of f are changing. It’s negative until two, while the slopes of f keep going down, and positive after two, while they go back up.

Read the hidden shape

Back at the telescope, the computer has to work the other way around. It never sees the blue curve, only the orange one, so let’s hide f and find out how much of it we can rebuild.

The first thing I ask the orange graph is whether it’s above zero or below it, because above zero the slopes are positive and the light is climbing, and below zero the light is falling. That already tells us where the peaks and valleys are. At x equals one, f prime crosses from positive to negative, so the light climbs and then falls, and that makes a crest. At three it crosses from negative to positive, which makes a trough.

That’s the first derivative test from last time, read straight off the graph of f prime. The second thing I ask is whether the orange graph itself is heading down or heading up. From zero to two it’s heading down, so the slopes of the light keep getting smaller. A curve whose slopes keep going down bends downward, which means it’s concave down, just like you saw last lesson.

From two to four f prime heads up, so the slopes keep going up and the light bends upward, which makes it concave up. The bend happens right at two, where the orange graph bottoms out, and that bend is an inflection point. Let’s switch the real light back on and check. Look at that, a crest at one, a bend at two and a trough at three, all from a graph that never showed us a single height.

Those two questions fit in one little table. Whether f prime is positive or negative tells you if f is increasing or decreasing, and whether f prime is rising or falling tells you if f is concave up or concave down. The square that trips people up is this one, where f prime is negative but rising. The slopes go negative three, negative two, negative one, so f is still heading downhill, it’s just flattening out as it goes, which makes it decreasing and concave up.

Let’s try a different square. On some interval, f prime is positive and decreasing, so which description fits the graph of f on that interval? You can even hear that square on the news. In 1969 President Nixon told the country that prices were still going up, but that the rates of increase were actually down, and that’s a price graph that’s increasing and concave down.

Now it’s your turn to be the computer. Here’s a different slice of light, and the only thing we know about it is this orange graph of its slopes. Drag the marker to the spot where the hidden light reaches its highest point, and let go when you think you have it.

What slopes can’t see

There’s one thing the slopes can never tell us, so watch the dots while I slide the whole slice of light upward. Not a single one of them moves. Sliding the light up or down doesn’t change a single slope, so the sensor is completely blind to it. Astronomers call that shift piston, and on paper it’s just adding a constant, plus C.

Luckily the telescope doesn’t care, because moving all the light together doesn’t blur anything. The computer bends the mirror into the opposite shape, the dent cancels out, and the star snaps back into a point.

On paper it looks like this. f, f plus two and f minus one all have the same derivative, this one orange curve, because adding a constant never changes a slope. That means f prime gives you the shape, and you need one actual point on f to pin down the height.

Suppose g has exactly the same derivative as our f, and g of zero equals five. Our f has f of zero equal to zero and f of three equal to zero, so what is g of three?

When zero isn’t enough

You met two traps in earlier lessons, and they come back every time you read a graph of f prime. Take f of x equals x cubed. Its derivative is three x squared, which equals zero right at x equals zero. It’s tempting to stop right there and call that a maximum or a minimum. But f prime is positive on both sides of zero, so x cubed climbs, flattens out for an instant, and then keeps right on climbing.

A zero of f prime is only a candidate. f prime changing from positive to negative gives you a maximum, and from negative to positive a minimum, but this f prime doesn’t change sign at all. Its concavity does flip from down to up, though, so x cubed has an inflection point at zero.

The same trap shows up one level down, with last lesson’s x to the fourth. Its f double prime is twelve x squared, which is zero at zero. But it’s positive on both sides, so x to the fourth is concave up the whole way and has no inflection point at all.

Whichever derivative you’re reading, a zero is only a candidate. What settles it is a sign change, so f prime changing sign gives you a peak or a valley, and f double prime changing sign gives you a bend. The graph of f prime touches the x-axis at x equals two and stays positive on both sides of it, so what does f have at x equals two?