Slope of a Line Between Two Points
The slope of a line between two given points is equal to where "rise" and "run" mean the same things they did when we first learned about the slopes of lines.Sample ProblemConsider the line drawn...
Slope of a Line Between Two Points on a Function
We've already done this in the case where the function is a line, but what happens if the line isn't straight? What if it's snake-shaped or U-shaped? What if it's as curvy as a mudflap? Now we'll f...
Slope at One Point?
The slope of the function f at the point (a, f(a)) is the limit of the slope of the secant line between x = a and x = a + h as h gets closer to 0. Whew, that's a mouthful.From the...
Estimating Derivatives from Tables
We can also estimate the derivative of a function f at a point a if we're given a table of values for f, but not given a formula. A table is a great way to record what a function does as we let h g...
Finding Derivatives Using Formulas
In this section we need to find derivatives "analytically," also known as "using the limit definition."Be Careful: "Find the derivative using the limit definition" does not mean estimating the de...
Average Rate of Change
A rate is a value that expresses how one quantity changes with respect to another quantity. For example, a rate in "miles per hour" expresses the increase in distance with respect to the number of...
Units, Words, and Notation
The phrase "instantaneous rate of change of f with respect to x at a" is a mouthful, because there are a lot of things we need to specify in order to be precise. This is why scientists sound like t...
How Tangent Lines Look
The tangent line to f at a is the line approached by the secant lines between a and a + h as h approaches 0.Tangent lines and secant lines are (usually) different things.A secant line must hit two...
When Tangent Lines Don't Exist
Tangent lines don't always exist. Since the slope of the tangent line to f at a is the same thing as the derivative of f at a, if f ' (a) doesn't exist then we can't draw a tangent line to f...
Tangent Lines and Derivatives
The following phrases all mean the same thing:the slope of f at athe derivative of f at a f ' (a)the slope of the tangent line to f at athe instantaneous rate of change of f at aSince f '...
Finding Tangent Lines
If we remember two things, we can write the equation for the tangent line to f at a given the formula for f and the value a where we want the tangent line to go.We can calculate f(a). ...
Differentiability and Continuity
We've had all sorts of practice with continuous functions and derivatives. Now it's time to see if these two ideas are related, if at all.We say a function is differentiable at a if f ' (a) ex...
The Derivative Function
The "derivative of f at a," written f ' (a), is a number that is equal to the slope of the function f at a.For any differentiable function f there is another function, known as the derivative...
Graphs of f ( x ) and f ' ( x )
From a graph of a function f(x) we can sketch graph of its derivative f ' (x). To do this, we use some things we talked about earlier.If f is decreasing, its slope (and hence its derivative)...
Rolle's Theorem
Rolle's Theorem says:Let f be a function that is continuous on the closed interval [a, b] is differentiable on the open interval (a, b), and has f (a) = f...
The Mean Value Theorem
After arriving at our Aunt Petunia's house for Thansgiving, we figured out that our average speed for the trip was 60 mph. Before diving into the sweet potato casserole, though, we asked ourselves...