Looking at Slopes

Welcome to Calculus!

One of our first concerns in calculus is determining the slopes of graphs. In algebra, you've calculated the slopes of lines, a special type of graph where the slope is the same everywhere. But we can also talk about how steep other graphs are.

Take a look at the graph of f(x) = x² below:

• For negative values of x (to the left of the graph), it's sloped downwards. The more negative x is, the steeper the slope.
• The graph "levels off" at x = 0. For values of x close to 0, the graph is close to horizontal.
• For positive values of x (to the right of the graph), it's sloped upwards. The larger x is, the steeper the slope.

In this interactive, I'll show you how to make formal sense of this notion, and we'll explore various reasons why people are interested in the slopes of graphs.

How do we Measure Slopes?

The slope of a line tells us about what angle it's at, but slopes aren't measured in degrees or radians. If we're going to be able to make sense of slopes, we need to be able to convert back and forth between numbers and what those slopes look like. You've learned previously how to calculate a slope by finding two points on your line and and calculating the rise/run = . But you don't need to calculate an exact slope to get a general sense of what's going on with a function.

• Fill out the table below. You may find it helpful to use the provided interactive: choose a slope and it will draw a line with that slope.

Zooming In

If you've played around with a graphing calculator, you may have noticed that when you zoom in on a graph, it tends to wind up looking like a line. Use the graphing calculator below to graph a function and zoom in on the graph.

• Type a formula into the text box and click the graph button
• You'll see a preview of the formula you typed in to the right of the graph button
• Zoom in with the mouse wheel or the buttons below the graph.
  • Try a few examples: how often does the graph wind up looking like a line?
  • Can you come up with a graph where you can zoom in as much as you want and it doesn't look like a line?

In algebra, you learned how to determine the y value of a point on a graph given an x value. The question we're interested in calculus is: Given a point on a graph, what is the slope of the line we get when we zoom in?

This is what we mean by the slope of a graph at a particular point. Just like with lines, this slope can tell us if a function is increasing or decreasing and how fast.

A Pattern in the Slopes

• Look at the graph y = x^2 below and fill out the table below for any 5 different points on the graph: (hint: stick to small, integer values of x). Remember that the slope of the graph is the slope of the line you get when you zoom in.
• Can you identify a pattern?

• Can you identify a pattern?

We'll learn why this pattern holds later on in calculus. If you're interested, try constructing a similar table for the following functions to see if you can figure out their patterns:

• 2x^2
• x^3 hint: try (x^3)/3
• (x^4)/4
• ln(x) hint: try x=1,2,3,4
• sin(x) hint: try x=0, pi/2, pi, 3pi/2, etc
• e^x

Linear Approximations

It's often handy to be able to work out the equation for the line you get when you zoom in: it gives us a line that closely resembles our graph when we zoom in really close, and lines are much easier to work with than some complicated curve.

Based on the values you entered above, work out the formula for each of the lines you saw. As you enter these formulas, this interactive will graph them.

(hint: the formula for a line with slope m passing through the point (p,q) is y = m(x-p)+q)

How are the lines and original graph related to each other? Be sure to zoom in and out on the graph to get an intuition for how these graphs are related.

Because these lines usually just touch the graph at a point, they're also called tangent lines.

Applications of Linear Approximations

Reasoning about Curved Surfaces:

We understand how flat (linear) mirrors work, and can use a bunch of linear approximations to approximate a curved mirror.

Calculating Numerical Approximations:

• Shown below is the graph of f(x) = sqrt(x) and the linear approximation to it at x = 4 (call this graph y = L(x)).
• Compare the y values of the linear approximation and the y values on the graph at x = 3, x = 5, and x = 4.1.
• How close does the linear approximation come to the actual square root of these values?

Calculating Velocity:

A linear function y = mt + b (t for time instead of x for position) corresponds to an object moving at constant velocity m. For objects that don't move at constant velocity, the tangent line corresponds to an object that matches position and speed around a particular instant (think of an action movie where two cars match position and speed so someone can jump from one to the other).

• Shown below is the graph of height versus time of a ball thrown up into the air.
• Click the play button to the bottom left of the graph or move the red bar under the graph to watch the animation.
• Check the "Show Tangent Line" checkbox above the graph. This graph is for a ball moving at constant velocity.
• Watch the animation again. Notice that the balls match velocity at the instant they pass each other.