Calculus 1 - Derivatives, Limits & Integrals Flashcards

Question 1

Limit Definition of the Derivative

Accepted Answer

The derivative of a function $f(x)$ at $x = a$ is given by $f'(a) = \lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}$, representing the instantaneous rate of change.

Question 2

Power Rule for Differentiation

Accepted Answer

For any real constant $n$, the derivative of a variable raised to a power is $\frac{d}{dx}[x^n] = n x^{n-1}$.

Question 3

Product Rule

Accepted Answer

For differentiable functions $u(x)$ and $v(x)$, the derivative of their product is $\frac{d}{dx}[u \cdot v] = u'v + uv'$.

Question 4

Quotient Rule

Accepted Answer

For differentiable functions $u(x)$ and $v(x)$, the derivative of their quotient is $\frac{d}{dx}\left[\frac{u}{v}\right] = \frac{u'v - uv'}{v^2}$.

Question 5

Chain Rule

Accepted Answer

For composite functions, the derivative is $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$, or in Leibniz notation, $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$.

Question 6

Derivatives of $\sin(x)$ and $\cos(x)$

Accepted Answer

$\frac{d}{dx}[\sin(x)] = \cos(x)$ and $\frac{d}{dx}[\cos(x)] = -\sin(x)$.

Question 7

Derivatives of $	an(x)$ and $\sec(x)$

Accepted Answer

$\frac{d}{dx}[	an(x)] = \sec^2(x)$ and $\frac{d}{dx}[\sec(x)] = \sec(x)	an(x)$.

Question 8

Derivatives of $e^x$ and $\ln(x)$

Accepted Answer

$\frac{d}{dx}[e^x] = e^x$ and $\frac{d}{dx}[\ln(x)] = \frac{1}{x}$ for $x > 0$.

Question 9

Derivatives of $\arcsin(x)$ and $\arctan(x)$

Accepted Answer

$\frac{d}{dx}[\arcsin(x)] = \frac{1}{\sqrt{1-x^2}}$ and $\frac{d}{dx}[\arctan(x)] = \frac{1}{1+x^2}$.

Question 10

L'Hôpital's Rule

Accepted Answer

If a limit $\lim_{x 	o c} \frac{f(x)}{g(x)}$ produces an indeterminate form $\frac{0}{0}$ or $\frac{\pm\infty}{\pm\infty}$, then $\lim_{x 	o c} \frac{f(x)}{g(x)} = \lim_{x 	o c} \frac{f'(x)}{g'(x)}$, provided the latter limit exists.

Question 11

Squeeze Theorem

Accepted Answer

If $g(x) \le f(x) \le h(x)$ near $x = c$, and $\lim_{x 	o c} g(x) = \lim_{x 	o c} h(x) = L$, then $\lim_{x 	o c} f(x) = L$.

Question 12

Intermediate Value Theorem (IVT)

Accepted Answer

If $f$ is continuous on $[a, b]$ and $N$ is any value between $f(a)$ and $f(b)$, then there exists at least one $c \in (a, b)$ such that $f(c) = N$.

Question 13

Mean Value Theorem (MVT)

Accepted Answer

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

Question 14

Rolle's Theorem

Accepted Answer

A special case of the MVT: If $f$ is continuous on $[a, b]$, differentiable on $(a, b)$, and $f(a) = f(b)$, then there exists at least one $c \in (a, b)$ such that $f'(c) = 0$.

Question 15

Concavity and the Second Derivative Test

Accepted Answer

If $f''(x) > 0$, $f$ is concave up. If $f''(x) < 0$, $f$ is concave down. If $f'(c) = 0$ and $f''(c) > 0$, $f(c)$ is a local minimum; if $f''(c) < 0$, $f(c)$ is a local maximum.

Question 16

Definition of a Definite Integral (Riemann Sum)

Accepted Answer

The definite integral is defined as $\int_{a}^{b} f(x) dx = \lim_{n 	o \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x$, where $\Delta x = \frac{b-a}{n}$.

Question 17

Fundamental Theorem of Calculus (Part 1)

Accepted Answer

If $f$ is continuous on $[a, b]$, then the area function $g(x) = \int_{a}^{x} f(t) dt$ is continuous on $[a, b]$, differentiable on $(a, b)$, and $g'(x) = f(x)$.

Question 18

Fundamental Theorem of Calculus (Part 2)

Accepted Answer

If $f$ is continuous on $[a, b]$, then $\int_{a}^{b} f(x) dx = F(b) - F(a)$, where $F$ is any antiderivative of $f$ (such that $F' = f$).

Question 19

Integration by $u$-Substitution

Accepted Answer

Reverses the chain rule: If $u = g(x)$, then $du = g'(x)dx$, transforming $\int f(g(x))g'(x) dx$ into the simpler form $\int f(u) du$.

Question 20

Power Rule for Integration

Accepted Answer

For $n 
eq -1$, $\int x^n dx = \frac{x^{n+1}}{n+1} + C$. For $n = -1$, $\int \frac{1}{x} dx = \ln|x| + C$.

Calculus 1 - Derivatives, Limits & Integrals

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