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Questions and Answers
Explain the Two-point Form of a Line and provide an example.
Explain the Two-point Form of a Line and provide an example.
The Two-point Form of a Line is given by $y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)$, where $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line. An example of the Two-point Form of a Line is $y - 3 = \frac{5 - 3}{4 - 2}(x - 2)$.
What is the geometrical meaning of derivatives and how is it related to the slope of a line?
What is the geometrical meaning of derivatives and how is it related to the slope of a line?
The geometrical meaning of derivatives is the slope of the tangent line to the graph of a function at a given point. This is related to the slope of a line as the derivative of a function at a point gives the slope of the tangent line to the graph of the function at that point.
Define the derivative of a function using the First Principle and provide an example.
Define the derivative of a function using the First Principle and provide an example.
The derivative of a function using the First Principle is defined as $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$. An example is finding the derivative of $f(x) = 3x^2 - 2x + 5$ using the First Principle.
What are the rules of differentiation and how are they applied to find the derivatives of standard functions?
What are the rules of differentiation and how are they applied to find the derivatives of standard functions?
Explain the Chain Rule for derivatives and provide a simple example.
Explain the Chain Rule for derivatives and provide a simple example.
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Study Notes
Two-point Form of a Line
- Represents a linear equation using two distinct points in a Cartesian plane.
- Formula: ( y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1) ), where ( (x_1, y_1) ) and ( (x_2, y_2) ) are the given points.
- Example: Points (2, 3) and (4, 7) yield ( y - 3 = 2(x - 2) ), simplifying to ( y = 2x - 1 ).
Geometrical Meaning of Derivatives
- Derivative indicates the rate of change of a function with respect to a variable.
- Represents the slope of the tangent line to the curve at a specific point.
- Positive derivative implies an increasing function, while a negative derivative indicates a decreasing function.
Derivative of a Function Using the First Principle
- First Principle defines the derivative as the limit of the average rate of change as the interval approaches zero.
- Formula: ( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} ).
- Example: For ( f(x) = x^2 ), ( f'(x) = \lim_{h \to 0} \frac{(x+h)^2 - x^2}{h} = \lim_{h \to 0} \frac{2xh + h^2}{h} = 2x ).
Rules of Differentiation
- Basic rules include Power Rule, Product Rule, Quotient Rule, and Chain Rule.
- Power Rule: ( \frac{d}{dx}(x^n) = n \cdot x^{n-1} ).
- Product Rule: ( \frac{d}{dx}(uv) = u'v + uv' ) for functions ( u ) and ( v ).
- Quotient Rule: ( \frac{d}{dx}(\frac{u}{v}) = \frac{u'v - uv'}{v^2} ).
Chain Rule for Derivatives
- Used to differentiate composite functions.
- Formula: ( \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x) ).
- Example: For ( f(x) = (3x + 2)^4 ), let ( g(x) = 3x + 2 ). Using the Chain Rule, the derivative is ( 4(3x + 2)^3 \cdot 3 = 12(3x + 2)^3 ).
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