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Questions and Answers
In numerical differentiation, what does the small number 'h' represent?
In numerical differentiation, what does the small number 'h' represent?
- The derivative of the function
- A small change in x (correct)
- The tangent of the function
- The limit of the function
What is the slope of the secant line through the points $(x, f(x))$ and $(x + h, f(x + h))$?
What is the slope of the secant line through the points $(x, f(x))$ and $(x + h, f(x + h))$?
- $f(x+h)+f(x)$
- ${f(x+h)-f(x) \over h}$ (correct)
- $f(x)$
- $f(x+h)-f(x)$
What is the slope of the secant line as 'h' approaches zero?
What is the slope of the secant line as 'h' approaches zero?
- The derivative of the function at x
- Infinity
- The slope of the tangent line (correct)
- Zero
What is the true derivative of $f$ at $x$?
What is the true derivative of $f$ at $x$?
What is another name for Newton's difference quotient?
What is another name for Newton's difference quotient?
Flashcards
What does 'h' represent?
What does 'h' represent?
A small change in the value of x, used to approximate derivatives.
Slope of the secant line?
Slope of the secant line?
The slope of the secant line is the difference in y-values divided by the difference in x-values: rise over run.
Limit as h approaches zero?
Limit as h approaches zero?
As 'h' gets infinitesimally small, the secant line approaches the tangent line.
What is the 'true derivative'?
What is the 'true derivative'?
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Newton's difference quotient?
Newton's difference quotient?
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Study Notes
Numerical Differentiation
- The small number 'h' represents a small increment added to the variable 'x' to approximate the change in the function's value.
- It is used to compute the difference quotient, which approximates the derivative.
Slope of the Secant Line
- The slope of the secant line through the points ((x, f(x))) and ((x + h, f(x + h))) is given by the formula (\frac{f(x + h) - f(x)}{h}).
- This slope estimates the average rate of change of the function (f) over the interval ([x, x + h]).
Slope as 'h' Approaches Zero
- As 'h' approaches zero, the slope of the secant line approaches the slope of the tangent line at point (x).
- This limit is referred to as the derivative of the function at that point.
True Derivative of (f) at (x)
- The true derivative of (f) at (x) is denoted as (f'(x)).
- It provides the instantaneous rate of change of the function at the specific point (x).
Newton's Difference Quotient
- Another name for Newton's difference quotient is the "first difference."
- It is used in numerical differentiation to approximate derivatives, particularly in Newton's method for root-finding.
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