What does the first derivative tell us about the graph of a function?
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What does the first derivative tell us about the graph of a function?
The first derivative of a function is an expression which tells us the slope of a tangent line to the curve at any instant. Because of this definition, the first derivative of a function tells us much about the function. If is positive, then must be increasing. If is negative, then must be decreasing.
Does the derivative give you the slope of the tangent line?
The derivative of a function gives us the slope of the line tangent to the function at any point on the graph. This can be used to find the equation of that tangent line.
What does the first derivative test tell you?
The first-derivative test examines a function’s monotonic properties (where the function is increasing or decreasing), focusing on a particular point in its domain. If the function “switches” from increasing to decreasing at the point, then the function will achieve a highest value at that point.
How do you find the derivative given a tangent line?
Now let’s look at an example function, f(x) = x3 + 3×2 + 1. We’ll find the tangent lines at a few different points. First of all, find the derivative: f ‘(x) = 3×2 + 6x….Example: A Polynomial.
| x | Tangent line, point-slope form | Tangent line, simplified |
|---|---|---|
| 0 | y = 0(x – 0) + 1 | y = 1 |
| 1 | y = 9(x – 1) + 5 | y = 9x – 4 |
How do you find slope with first derivative?
1) Find the first derivative of f(x). 2) Plug x value of the indicated point into f ‘(x) to find the slope at x. 3) Plug x value into f(x) to find the y coordinate of the tangent point. 4) Combine the slope from step 2 and point from step 3 using the point-slope formula to find the equation for the tangent line.
How does the derivative give the slope?
When you plug in an x-value into a function’s derivative, the y-values you get back FROM THE DERIVATIVE tell you the slope of a tangent line to the original function at that value of x.
What is the first derivative of the tangent line?
The first derivative can be interpreted as an instantaneous rate of change. The first derivative can also be interpreted as the slope of the tangent line. The Derivative as the Slope of a Tangent Line Recall that the definition of the derivative is
What does the first derivative of a function tell you?
Quick Overview 1 The first derivative primarily tells us about the direction the function is going. That is, it tells us if the function is increasing or decreasing. 2 The first derivative can be interpreted as an instantaneous rate of change. 3 The first derivative can also be interpreted as the slope of the tangent line.
What does the sign of the derivative tell us?
The sign of the derivative at a particular point will tell us if the function is increasing or decreasing near that point. This is readily apparent when we think of the derivative as the slope of the tangent line. As shown in the two graphs below, when the slope of the tangent line is positive, the function will be increasing at that point.
What is the difference between the first and second derivative?
While the first derivative can tell us if the function is increasing or decreasing, the second derivative tells us if the first derivative is increasing or decreasing. If the second derivative is positive, then the first