Mathematics Calculus Syllabus

State L'Hopital's Rule and the indeterminate forms it applies to.

If $\lim \frac{f(x)}{g(x)}$ is of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then $\lim \frac{f(x)}{g(x)} = \lim \frac{f'(x)}{g'(x)}$, provided the latter limit exists.

What is the geometric meaning of the derivative $f'(a)$?

It is the slope of the tangent line to the graph of $f$ at the point $(a, f(a))$; equivalently the instantaneous rate of change of $f$ at $a$.

State the Mean Value Theorem (MVT).

If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists $c \in (a,b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$.

State Rolle's Theorem.

If $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a) = f(b)$, then there exists $c \in (a,b)$ with $f'(c) = 0$.

How do you find critical points of a function $f$?

Critical points are values $x$ in the domain where $f'(x) = 0$ or $f'(x)$ does not exist.

State the First Derivative Test for local extrema.

At a critical point $c$: if $f'$ changes from $+$ to $-$, $f$ has a local maximum; if $f'$ changes from $-$ to $+$, a local minimum; if no sign change, neither.

State the Second Derivative Test for a critical point $c$ where $f'(c)=0$.

If $f''(c) > 0$, $f$ has a local minimum at $c$; if $f''(c) < 0$, a local maximum; if $f''(c) = 0$, the test is inconclusive.

What does the sign of the second derivative $f''(x)$ tell you about concavity?

If $f''(x) > 0$ the graph is concave up; if $f''(x) < 0$ it is concave down.

What is an inflection point?

A point on the graph where the concavity changes sign (from concave up to down or vice versa); $f''$ is zero or undefined there and changes sign.

Define an antiderivative of a function $f$.

A function $F$ is an antiderivative of $f$ if $F'(x) = f(x)$. All antiderivatives differ by a constant: $F(x) + C$.

What is the difference between a definite and an indefinite integral?

An indefinite integral $\int f(x)\,dx$ gives a family of antiderivatives (a function $+\,C$); a definite integral $\int_a^b f(x)\,dx$ gives a number (net signed area).

State the method of $u$-substitution for integrals.

Let $u = g(x)$ so $du = g'(x)\,dx$, then $\int f(g(x))g'(x)\,dx = \int f(u)\,du$. For definite integrals, change limits accordingly.

What is the linearity property of integration?

$\int \big[a f(x) + b g(x)\big]\,dx = a\int f(x)\,dx + b\int g(x)\,dx$ for constants $a, b$.

How do you compute the area between two curves $f(x) \geq g(x)$ on $[a,b]$?

$$A = \int_a^b \big[f(x) - g(x)\big]\,dx$$

State the disk method for the volume of a solid of revolution about the $x$-axis.

$$V = \pi \int_a^b [f(x)]^{2}\,dx$$

State the shell method for volume of revolution about the $y$-axis.

$$V = 2\pi \int_a^b x\,f(x)\,dx$$

What is the formula for the arc length of $y = f(x)$ from $x=a$ to $x=b$?

$$L = \int_a^b \sqrt{1 + [f'(x)]^{2}}\,dx$$

What is the average value of a continuous function $f$ on $[a,b]$?

$$f_{\text{avg}} = \frac{1}{b-a}\int_a^b f(x)\,dx$$

What distinguishes a removable, jump, and infinite discontinuity?

Removable: limit exists but $\neq f(a)$ (hole). Jump: left and right limits exist but differ. Infinite: a one-sided limit is $\pm\infty$ (vertical asymptote).

What is the difference between a local (relative) extremum and a global (absolute) extremum?

A local extremum is the largest/smallest value in a neighborhood; a global extremum is the largest/smallest over the entire domain or interval.

How do you find the absolute extrema of a continuous $f$ on a closed interval $[a,b]$ (Closed Interval Method)?

Evaluate $f$ at all critical points in $(a,b)$ and at the endpoints $a$ and $b$; the largest value is the absolute max and the smallest is the absolute min.

What is implicit differentiation used for?

To find $\frac{dy}{dx}$ when $y$ is defined implicitly by an equation; differentiate both sides with respect to $x$, treating $y$ as a function of $x$ (using the chain rule), then solve for $\frac{dy}{dx}$.

What is the linear (tangent line) approximation of $f$ near $x = a$?

$$L(x) = f(a) + f'(a)(x - a)$$

State the Squeeze (Sandwich) Theorem.

If $g(x) \leq f(x) \leq h(x)$ near $a$ and $\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$, then $\lim_{x \to a} f(x) = L$.

What is the value of $\lim_{x \to 0} \frac{\sin x}{x}$?

$\lim_{x \to 0} \frac{\sin x}{x} = 1$.

What conditions define a vertical asymptote versus a horizontal asymptote?

Vertical asymptote at $x = a$: $\lim_{x \to a} f(x) = \pm\infty$. Horizontal asymptote at $y = L$: $\lim_{x \to \pm\infty} f(x) = L$.

What is the relationship between differentiability and continuity?

If $f$ is differentiable at $a$, then it is continuous at $a$. The converse is false: continuity does not imply differentiability (e.g., $|x|$ at $0$).

What is the Taylor series of $f$ centered at $a$?

$$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^{n}$$ A Maclaurin series is the special case $a = 0$.

State the Integral Test for convergence of a series.

If $f$ is continuous, positive, and decreasing for $x \geq 1$ with $a_n = f(n)$, then $\sum_{n=1}^{\infty} a_n$ and $\int_1^{\infty} f(x)\,dx$ both converge or both diverge.

For what values of $p$ does the $p$-series $\sum_{n=1}^{\infty} \frac{1}{n^{p}}$ converge?

It converges if $p > 1$ and diverges if $p \leq 1$.