Mathematics Calculus Flashcards

24 sample cards from the Calculus deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. 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.

2. 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.

3. 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.

4. 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: $F(x) + C$.

5. 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).

6. 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.

7. 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 $a, b$.

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

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

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

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

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

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

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

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

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

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

13. 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).

14. 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.

15. 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.

16. 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 $\frac{dy}{dx}$.

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

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

18. 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 $\lim_{x \to a} f(x) = L$.

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

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

20. 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 $y = L$: $\lim_{x \to \pm\infty} f(x) = L$.

21. 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 $0$).

22. 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 $a = 0$.

23. 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.

24. 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 $p \leq 1$.

What this deck covers

Calculus flashcards FAQ