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Derivatives
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Calculus
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Derivatives Examples
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. What happens if the line isn't straight? What if it's snakeshaped or Ushaped? What if it's as curvy as a mudflap? Now we'll find...
Estimating Derivatives Given the Formula
The derivative (or slope) of f at a is defined as Since f'(a) is defined as a limit, we immediately have one strategy for finding it:Pick a value of h that's close to 0. Find the slope...
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. Check out the examples and exercises to learn how.
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. This applet lets us watch the secant line approach the tangent line as we drag the point...
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). W...
Using Tangent Lines to Approximate Function Values
"Approximation" is what we do when we can't or don't want to find an exact value. We're going to approximate actual function values using tangent lines.We pointed out earlier that if we zoom in fa...
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 make a sketched 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 der...
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
The Mean Value Theorem is a glorified version of Rolle's Theorem. The Mean Value Theorem states: If f is continuous on [a, b], and f is differentiable on (a, b), then th...
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