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) = x², x = 1

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First, understand that the expression limh→0 (f(x+h) − f(x)) / h represents the derivative of the function f(x) at a specific point x. This is the definition of the derivative using the limit process.

Identify the function f(x) = x² and the point x = 1 where you need to evaluate the limit. This means you are finding the derivative of f(x) = x² at x = 1.

Substitute f(x) = x² into the limit expression: limh→0 ((1+h)² − 1²) / h. This involves substituting x = 1 into the function and considering the increment h.

Expand the expression (1+h)² to get 1 + 2h + h². This is a crucial step to simplify the expression inside the limit.

Simplify the expression: ((1 + 2h + h²) − 1) / h, which reduces to (2h + h²) / h. Then, factor out h from the numerator to get h(2 + h) / h, and cancel h from the numerator and denominator, leaving 2 + h. Finally, evaluate the limit as h approaches 0, which gives the derivative at x = 1.

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Limits

In calculus, a limit describes the value that a function approaches as the input approaches a certain point. It is fundamental for understanding continuity, derivatives, and integrals. The notation lim h→0 indicates that we are examining the behavior of a function as the variable h approaches zero, which is crucial for defining instantaneous rates of change.

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 (the slope of the secant line) as the interval approaches zero. For the function f(x) = x², the derivative at x = 1 can be found using the limit definition, yielding the slope of the tangent line at that point.

Secant and Tangent Lines

A secant line intersects a curve at two or more points, providing an average rate of change between those points. In contrast, a tangent line touches the curve at a single point and represents the instantaneous rate of change at that point. The relationship between secant and tangent lines is essential in calculus, as the limit of the secant line's slope as the two points converge gives the derivative.

Limits of Average Rates of ChangeBecause 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) = x², x = 1