8.1.1 An Introduction to Curve Sketching

Question 1

Introduction to Curve Sketching

Answer 1

• 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.

Question 2

note

Answer 2

• 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.

Question 3

How can you tell if a graph is symmetric about the y‑axis?

Answer 3

If f(−x)=f(x), then the graph is symmetric about the y-axis.

Question 4

How can you tell if a graph is symmetric about the origin?

Answer 4

If f (x) = − f (−x), then the graph is symmetric about the origin.

Question 5

How can you tell that a curve doesn’t wiggle between plotted points?

Answer 5

By analyzing the slope as it changes.

Question 6

Which of the following figures is not symmetric across both the origin and the y‑axis?

Answer 6

The line y = x

Question 7

In which application is the derivative not used?

Answer 7

Calculating area under a curve