Slopes of tangent lines For the following functions, make a...

Slopes of tangent lines For the following functions, make a table of slopes of secant lines and make a conjecture about the slope of the tangent line at the indicated point.
$$f(x)=e^{x} \quad \text { at } x=0$$

Key Concepts

Limit Process

The limit process involves determining the value that a function or expression approaches as the input approaches some value. In the context of slopes and tangent lines, taking the limit of the difference quotient as the interval shrinks to zero leads to the concept of the derivative, capturing the instantaneous rate of change of the function.

Difference Quotient

The difference quotient is the expression that calculates the average rate of change of a function over an interval. It is defined as (f(x + h) - f(x)) / h, and as h approaches zero, this quotient approaches the derivative, which is the slope of the tangent line.

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point and has the same instantaneous rate of change as the curve does there. It represents the best linear approximation to the function at that point, capturing the direction in which the function is headed.

Secant Line

A secant line is a line that intersects a curve at two distinct points. The slope of a secant line between two points on a function gives the average rate of change of the function over that interval. As the two points get closer together, the secant slope approaches the slope of the tangent line.

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