Final answer:
The tangent line to a curve in multivariable calculus represents the instantaneous rate of change of the curve at a specific point. Option (D) is correct.
Step-by-step explanation:
The tangent line to a curve in multivariable calculus represents the instantaneous rate of change of the curve at a specific point.
When we take the tangent line at a point on a curve, the slope of that tangent line gives us the rate at which the curve is changing at that point. This rate of change is known as the instantaneous rate of change.
Therefore, the correct answer is b) Instantaneous rate of change.
For a curve parametrized by c(t), the derivative c′(t) is a vector that is tangent to the curve. We can use this fact to derive an equation for a line tangent to the curve. Fix a time t0. The line through point c(t0) in the direction parallel to the tangent vector c′(t0) will be a tangent line to the curve.
The tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve.