Calc 1 Section 4-5: Curve Sketching

Welcome back to Professor Baker's Math Class! In this section, we'll be exploring the wonderful world of curve sketching. Mastering these techniques will allow you to visualize and understand the behavior of functions in a whole new way. Let's dive in!

A. Domain

It's always useful to start by determining the domain $D$ of $f$, that is, the set of values of $x$ for which $f(x)$ is defined. Keep an eye out for:

  • Fractions: Avoid division by zero. For example, in $f(x) = \frac{1}{x}$, $x \neq 0$.
  • Even Roots: Ensure the radicand (the expression inside the root) is non-negative. For example, in $f(x) = \sqrt{x}$, $x \geq 0$.
  • Logarithms: The argument of a logarithm must be positive. For example, in $f(x) = \ln(x)$, $x > 0$.
  • Real-life restrictions.

B. Intercepts

  • The y-intercept is $f(0)$, telling us where the curve intersects the y-axis.
  • To find x-intercepts, set $y = 0$ and solve for $x$. This can sometimes be tricky, so don't worry if it's difficult.

Every function has at most one y-intercept.

C. Symmetry

  • Even Functions: If $f(-x) = f(x)$ for all $x$ in $D$, then $f$ is an even function, and the curve is symmetric about the y-axis. Examples: $y = x^2$, $y = |x|$.
  • Odd Functions: If $f(-x) = -f(x)$ for all $x$ in $D$, then $f$ is an odd function, and the curve is symmetric about the origin. Examples: $y = x$, $y = x^3$, $y = \sin x$.
  • Periodic Functions: If $f(x + p) = f(x)$ for all $x$ in $D$, where $p$ is a positive constant, then $f$ is a periodic function, and $p$ is the period. Example: $y = \sin x$ has a period of $2\pi$.

D. Asymptotes

  • Horizontal Asymptotes: If $\lim_{x \to \infty} f(x) = L$ or $\lim_{x \to -\infty} f(x) = L$, then $y = L$ is a horizontal asymptote.
  • Vertical Asymptotes: The line $x = a$ is a vertical asymptote if at least one of the following is true: $\lim_{x \to a^+} f(x) = \pm \infty$ or $\lim_{x \to a^-} f(x) = \pm \infty$

E. Intervals of Increase or Decrease

Use the I/D Test: Compute $f'(x)$ and find the intervals on which $f'(x)$ is positive (f is increasing) and the intervals on which $f'(x)$ is negative (f is decreasing).

F. Local Maximum or Minimum Values

Find the critical numbers of $f$ (where $f'(x) = 0$ or $f'(x)$ does not exist). Use the First Derivative Test. If $f'$ changes from positive to negative at a critical number $c$, then $f(c)$ is a local maximum. If $f'$ changes from negative to positive at $c$, then $f(c)$ is a local minimum.

G. Concavity and Points of Inflection

Compute $f''(x)$ and use the Concavity Test. The curve is concave upward where $f''(x) > 0$ and concave downward where $f''(x) < 0$. Inflection points occur where the direction of concavity changes.

Keep practicing, and you'll become a curve-sketching pro in no time!