Limits of Average Rates of Change

Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form limh→0 (f(x+h) − f(x)) / h occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function f.

f(x) = 3x - 4, x = 2

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Identify the function f(x) = 3x - 4 and the point x = 2 where we need to evaluate the limit.

Substitute f(x) and f(x+h) into the limit expression: lim(h→0) [(f(x+h) - f(x)) / h]. This becomes lim(h→0) [(3(x+h) - 4) - (3x - 4)] / h.

Simplify the expression inside the limit: (3(x+h) - 4) - (3x - 4) simplifies to 3x + 3h - 4 - 3x + 4, which further simplifies to 3h.

The limit expression now becomes lim(h→0) [3h / h]. Simplify this expression by canceling h in the numerator and denominator, resulting in lim(h→0) [3].

Since the expression is now a constant, the limit as h approaches 0 is simply the constant value, which is 3.

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Limits

In calculus, a limit is a fundamental concept that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near specific points, which is crucial for defining derivatives and integrals. The notation lim h→0 f(x+h) indicates examining the function f as the increment h approaches zero, allowing us to analyze instantaneous rates of change.

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Average Rate of Change

The average rate of change of a function over an interval is calculated as the change in the function's value divided by the change in the input value. Mathematically, it is expressed as (f(b) - f(a)) / (b - a) for the interval [a, b]. This concept is essential for understanding how a function behaves over an interval and serves as a foundation for the more precise instantaneous rate of change, which is explored through limits.

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Derivative

The derivative of a function at a point represents the instantaneous rate of change of the function with respect to its variable. It is defined as the limit of the average rate of change as the interval approaches zero, formally expressed as f'(x) = lim h→0 (f(x+h) - f(x)) / h. Derivatives are crucial in calculus for analyzing the behavior of functions, finding slopes of tangent lines, and solving optimization problems.

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