To determine if a function is continuous at a specific point, we need to check three conditions:
1. The limit of the function exists at that point.
2. The value of the function at that point exists.
3. The limit and the value of the function are equal.
Let's apply these conditions to your function, f(x) = {(x+1) if x>3, (x-3) if x<=3}, to check if it is continuous at x = 3.
1. To find the limit of the function as x approaches 3, we need to find the one-sided limits from both directions:
- As x approaches 3 from the right (x > 3), the function becomes f(x) = (x + 1). So, the limit as x approaches 3 from the right is 4.
- As x approaches 3 from the left (x <= 3), the function becomes f(x) = (x - 3). So, the limit as x approaches 3 from the left is 0.
2. To find the value of the function at x = 3, we need to evaluate it based on the given conditions:
- Since x = 3 is not greater than 3, the function becomes f(3) = (3 - 3) = 0.
3. Now, we compare the limit and the value of the function at x = 3:
- The limit as x approaches 3 from the right is 4, and the value of the function at x = 3 is 0. Therefore, the limit and the value are not equal.
Since the limit and the value of the function are not equal at x = 3, f(x) is not continuous at x = 3.
1. The limit of the function exists at that point.
2. The value of the function at that point exists.
3. The limit and the value of the function are equal.
Let's apply these conditions to your function, f(x) = {(x+1) if x>3, (x-3) if x<=3}, to check if it is continuous at x = 3.
1. To find the limit of the function as x approaches 3, we need to find the one-sided limits from both directions:
- As x approaches 3 from the right (x > 3), the function becomes f(x) = (x + 1). So, the limit as x approaches 3 from the right is 4.
- As x approaches 3 from the left (x <= 3), the function becomes f(x) = (x - 3). So, the limit as x approaches 3 from the left is 0.
2. To find the value of the function at x = 3, we need to evaluate it based on the given conditions:
- Since x = 3 is not greater than 3, the function becomes f(3) = (3 - 3) = 0.
3. Now, we compare the limit and the value of the function at x = 3:
- The limit as x approaches 3 from the right is 4, and the value of the function at x = 3 is 0. Therefore, the limit and the value are not equal.
Since the limit and the value of the function are not equal at x = 3, f(x) is not continuous at x = 3.
To determine if the function f(x) = (x-9)/√(x-3) is continuous at a specific point, we need to check three conditions:
1. The limit of the function exists at that point.
2. The value of the function at that point exists.
3. The limit and the value of the function are equal.
Let's consider the point x = 3 to determine if the function is continuous there:
1. To find the limit of the function as x approaches 3, we take the limit of f(x) as x approaches 3 from both the left and the right sides:
- As x approaches 3 from the left side (x < 3), we substitute x = 3 into the function and evaluate: lim(x→3-) (x-9)/√(x-3) = lim(x→3-) (3-9)/√(3-3) = -6/0, which is undefined.
- As x approaches 3 from the right side (x > 3), we substitute x = 3 into the function and evaluate: lim(x→3+) (x-9)/√(x-3) = lim(x→3+) (3-9)/√(3-3) = -6/0, which is also undefined.
Since both the left-sided and right-sided limits are undefined, the limit of the function does not exist at x = 3.
2. To find the value of the function at x = 3, we substitute x = 3 into the function: f(3) = (3-9)/√(3-3) = -6/0, which is undefined.
3. As the limit does not exist at x = 3, the condition of the limit and value being equal is not satisfied.
Therefore, we conclude that the function f(x) = (x-9)/√(x-3) is not continuous at x = 3.
To find the derivative of the function y = 3x^2 + 2 at x = 1, we can differentiate the function with respect to x using the power rule for derivatives.
The power rule states that if we have a function of the form y = ax^n, where a is a constant and n is any real number, then the derivative of this function with respect to x is given by dy/dx = nax^(n-1).
Using the power rule, we can differentiate the function y = 3x^2 + 2 term by term:
dy/dx = d/dx (3x^2) + d/dx (2)
Differentiating each term, we get:
dy/dx = 6x + 0
Simplifying, we have:
dy/dx = 6x
Now, to find the derivative at x = 1, we substitute x = 1 into the derivative:
dy/dx = 6(1) = 6
Therefore, the derivative of the function y = 3x^2 + 2 at x = 1 is 6.
To find the derivative of the function y = -5x, we need to differentiate the function with respect to x. Using the power rule for derivatives, we know that the derivative of a constant times x is simply the constant itself.
In this case, the function is y = -5x, and the derivative of -5x with respect to x is -5.
To find the derivative at x = -1, we substitute x = -1 into the derivative:
dy/dx = -5
Therefore, the derivative of the function y = -5x at x = -1 is -5.
The rule you are referring to is called the Power Rule. The Power Rule states that the derivative of a function of the form y = x^n, where n is any real number, is given by dy/dx = nx^(n-1). This rule allows us to find the derivative of a function when the variable is raised to a constant power.
is y^(1)=nx^(n-1)