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Definition: A function f is differentiable at a if f'(a) exists. it is differentiable on an open interval (a, b) [or (a, ∞) or (-∞, a) or (-∞, ∞)]if it is differentiable at every number in the interval.
Example: Where is the function f(x) = |x| differentiable?
Answer:
1. f is differentiable for any x > 0 and x < 0.
2. f is not differentiable at x = 0.
That's mean that the curve y = |x| has not a tangent at (0, 0).
Thus, both continiuty and differentiability are desirable properties for a function to have. These properties are related.
Theorem: If f is differentiable at a, then f is continuous at a.
The converse theorem is false, that is, there are functions that are continuous but not differentiable. (As we saw at the example above. f(x) = |x| is contionuous at 0, but is not differentiable at 0).
The three ways for f not to be differentiable at aare:
a) if the graph of a function f has a "corner" or a "kink" in it,
b) a discontinuity,
c) a vertical tangent
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How do you determine the relative minimum and relative maximum values of functions and the intervals on which functions are decreasing or increasing?
You take the derivative of the function. The derivative is another function that tells you the slope of the original function at any point. (If you don't know about derivatives already, you can learn the details on how to calculate in a calculus textbook. Or read the Wikipedia article for a brief introduction.) Once you have the derivative, you solve it for zero (derivative = 0). Any local maximum or minimum either has a derivative of zero, has no defined derivative, or is a border point (on the border of the interval you are considering). Now, as to the intervals where the function increase or decreases: Between any such maximum or minimum points, you take any random point and check whether the derivative is positive or negative. If it is positive, the function is increasing.
How do you find second derivative of a function?
All it means to take the second derivative is to take the derivative of a function twice. For example, say you start with the function y=x2+2x The first derivative would be 2x+2 But when you take the derivative the first derivative you get the second derivative which would be 2
What is the derivative of e2x-1?
3
What is the difference between nonlinear and linear functions?
Linear function:No variable appears in the function to any power other than 1.A periodic input produces no new frequencies in the output.The function's first derivative is a number; second derivative is zero.The graph of the function is a straight line.Non-linear function:A variable appears in the function to a power other than 1.A periodic function at the input produces new frequencies in the output.The function's first derivative is a function; second derivative is not zero.The graph of the function is not a straight line.
How do you find the slope of a tangent?
Take the derivative of the function.
If second derivative is 0 and third derivative is 0 What is true about that point?
If the second derivative of a function is zero, then the function has a constant slope, and that function is linear. Therefore, any point that belongs to that function lies on a line.
When does the derivative of a function exist at a given point?
Let f be a function and a be the given point you are considering. Then,f(x) - f(a)---------------(x-a)is the difference quotient. If the limit as x approaches a exists, then the function is differentiable at a, or we say the derivative exists at a. If that limit does not exist, then the derivative does not exist at that point.
What is differentiability in math?
A function is differentiable at a point if the derivative exists there.
What is the second derivative of a function's indefinite integral?
well, the second derivative is the derivative of the first derivative. so, the 2nd derivative of a function's indefinite integral is the derivative of the derivative of the function's indefinite integral. the derivative of a function's indefinite integral is the function, so the 2nd derivative of a function's indefinite integral is the derivative of the function.
How do you graph the slope of a function?
For example, if the slope at a certain point is 1.5, you can draw a line that goes through the specified point, with that slope. The line would represent the slope at that point. If you want to graph the slope at ALL POINTS, take the derivative of the function, and graph the derivative. The derivative shows the slope of a function at all points.
How do you find the minimum or maximum of a function?
By taking the derivative of the function. At the maximum or minimum of a function, the derivative is zero, or doesn't exist. And end-point of the domain where the function is defined may also be a maximum or minimum.
How do you Find derivative to get the slope?
the deivative of a function is the gradient, at a point if you can sub in the x coordinate for that point
How is the function differentiable in graph?
If the graph of the function is a continuous line then the function is differentiable. Also if the graph suddenly make a deviation at any point then the function is not differentiable at that point . The slope of a tangent at any point of the graph gives the derivative of the function at that point.
How do you determine the relative minimum and relative maximum values of functions and the intervals on which functions are decreasing or increasing?
You take the derivative of the function. The derivative is another function that tells you the slope of the original function at any point. (If you don't know about derivatives already, you can learn the details on how to calculate in a calculus textbook. Or read the Wikipedia article for a brief introduction.) Once you have the derivative, you solve it for zero (derivative = 0). Any local maximum or minimum either has a derivative of zero, has no defined derivative, or is a border point (on the border of the interval you are considering). Now, as to the intervals where the function increase or decreases: Between any such maximum or minimum points, you take any random point and check whether the derivative is positive or negative. If it is positive, the function is increasing.
what is a derivative of a function?
Another function, which at every point has the value of the slope of the original function. For example, the parabola y = x2 has, at any point, the slope 2x, so 2x is the derivative of the original function. The Wikipedia article - http://en.wikipedia.org/wiki/Derivative - gives a more detailed introduction, but if you want to really learn about derivatives, you should pick up textbook on introductory calculus.
How does one figure the rate of change?
The rate of change of a function is found by taking the derivative of the function. The equation for the derivative gives the rate of change at any point. This method is used frequently in calculus.
Why does taking the derivative of a function give you the slope of the tangent?
Why: Because that's what the derivative means, the way it is defined - the slope of the curve at any point of the line.
