Question

The (a) ____ of the (b) ____ to the graph of f(x) at the point with x=a is the limit of the slope of the (c) ____ through points (a, f(a)) and (b, f(b)) as (d) ____.

a) Tangent, line, secant, b approaches a
b) Slope, curve, tangent, b approaches a
c) Area, function, line, b approaches x
d) Derivative, axis, secant, b approaches infinity

Answer 1

The correct answers are: (a) Tangent, (b) curve, (c) secant, (d) b approaches a. This describes the process of finding the derivative, which is the slope of the tangent line at a specific point on the curve of a function.

Step-by-step explanation:

The answer to the fill-in-the-blank question is: (a) Tangent, (b) curve, (c) secant, (d) b approaches a. The definition of the slope of the tangent line to the graph of a function at a point involves calculus concepts. Specifically, the slope of the tangent line to the graph of function f(x) at the point with x=a is the limit of the slope of the secant line through points (a, f(a)) and (b, f(b)) as b approaches a. This concept is a foundational idea in differential calculus and is related to the derivative of a function. The derivative of a function at a point gives the instantaneous rate of change of the function at that point, which is the slope of the tangent line to the function's graph at that point.

Graphs showing lines with various slopes illustrate the effect of the slope value on the line's direction: upward sloping, horizontal, or downward sloping. Understanding the algebra of straight lines, including slope and y-intercept, is essential in learning how to find the tangent line's equation at a given point.