Flashcards in 8.1.1 An Introduction to Curve Sketching Deck (7):
Introduction to Curve Sketching
- Applications of the derivative include motion problems, linear approximations, optimization, related rates, and curve sketching.
- The techniques used in algebra for graphing functions do not demonstrate subtle behaviors of curves. You can use the derivative to describe the curvature of a graph more accurately.
- The derivative has many real-world applications,
ranging from motion problems to related rates.
- But the derivative is more powerful still. You can
use the derivative to more accurately sketch the
graph of a function.
- In algebra, you probably learned to sketch functions
by plotting points. Then you connected the points
together, assuming that they connected smoothly
and without any wiggles.
- However, you cannot be sure that graphs do not
wiggle with the naïve approach used in algebra.
Calculus will prove that the graph does not wiggle.
How can you tell if a graph is symmetric about the y‑axis?
If f(−x)=f(x), then the graph is symmetric about the y-axis.
How can you tell if a graph is symmetric about the origin?
If f (x) = − f (−x), then the graph is symmetric about the origin.
How can you tell that a curve doesn’t wiggle between plotted points?
By analyzing the slope as it changes.
Which of the following figures is not symmetric across both the origin and the y‑axis?
The line y = x