What is the formula for the gradient of the secant line between points P0(x0, f(x0)) and P1(x1, f(x1))?
Understand the Problem
The question is asking for the mathematical formula that defines the gradient of the secant line that connects the points P0 and P1 on a function f. The gradient, or slope, of the secant line is typically calculated as the change in the function values (the vertical change) divided by the change in the x-values (the horizontal change) between the two points.
Answer
The gradient of the secant line is given by $m = \frac{f(x_1) - f(x_0)}{x_1 - x_0}$.
Answer for screen readers
The gradient of the secant line connecting the points ( P_0 ) and ( P_1 ) on the function ( f ) is given by: $$ m = \frac{f(x_1) - f(x_0)}{x_1 - x_0} $$
Steps to Solve
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Identify the points and function Start with two points ( P_0(x_0, f(x_0)) ) and ( P_1(x_1, f(x_1)) ) on the function ( f ).
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Define the changes in values Calculate the change in the function values (vertical change) which is given by: $$ \Delta y = f(x_1) - f(x_0) $$
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Define the change in x-values Calculate the change in the x-values (horizontal change) as follows: $$ \Delta x = x_1 - x_0 $$
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Calculate the gradient Using the changes calculated above, the gradient (or slope) of the secant line can be expressed as: $$ m = \frac{\Delta y}{\Delta x} = \frac{f(x_1) - f(x_0)}{x_1 - x_0} $$
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Final formula for the slope of the secant line The final formula concisely capturing the gradient of the secant line connecting ( P_0 ) and ( P_1 ) is: $$ m = \frac{f(x_1) - f(x_0)}{x_1 - x_0} $$
The gradient of the secant line connecting the points ( P_0 ) and ( P_1 ) on the function ( f ) is given by: $$ m = \frac{f(x_1) - f(x_0)}{x_1 - x_0} $$
More Information
The slope of the secant line is a key concept in calculus that helps in approximating the slope of the tangent line at points on a function. It is foundational for understanding derivatives and rates of change.
Tips
- Confusing the secant line's slope with that of the tangent line; the secant line connects two distinct points, while the tangent line touches the curve at one point only.
- Forgetting to use function values for ( y ) when calculating ( \Delta y ); it should always be ( f(x_1) - f(x_0) ), not just ( x_1 - x_0 ).
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