[Calculus] Quick Review for Tutorial 7

1. Definition of Concavity
Let $f$ be a differentiable function defined on an open interval $D$. The graph of f is said to be

  • concave up (on $D$) if $f\prime$ is increasing on $D$;
  • concave down (on $D$) if $f\prime$ is decreasing on $D$.

2. Remark of Concavity
Whenever $f$ is twice differentiable on $D$:

  • if $f\prime\prime(x)>0$ for all $x\in D$, then the graph off is concave up;
  • if $f\prime\prime(x)<0$ for all $x\in D$, then the graph off is concave down.

3. Second Derivative Test
Theorem (Second Derivative Test)
Let $f$ be a function that is twice differentiable on an open interval $D$, and let $c$ be a critical point of $f$.

  • (i) If $f\prime\prime(c)>0$, then $f$ has a local minimum at $c$.
  • (ii) If $f \prime\prime(c) < 0$, then $f$ has a local maximum at $c$.

4. L’Hopital’s Rule
Let $c \in R$. Suppose that $f$ and $g$ are differentiable on $D := (c −a, c +a)-{c}$ for some $a > 0$, and that $g′(x) \neq 0$ for all $x \in D$. Suppose that one of the following two conditions holds:

  • $lim_{x\rightarrow c}f(x)=0= lim_{x\rightarrow c}g(x);$
  • $lim_{x\rightarrow c}f(x)\in{−∞,∞}$ and $lim_{x\rightarrow c}g(x)\in{−∞,∞}.$
    $$lim_{x\rightarrow c} {f(x)\over g(x)} = lim_{x\rightarrow c} {f\prime(x)\over g\prime(x)} $$

provided that the limit on the right exists (or is ∞ or −∞).

The code for Problem 4(https://octave-online.net):
t = [-2.5:0.01:2.5];
x = 2t - t.^2;
y = 3
t - t.^3;