Calculus Flashcards Flashcards

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Limit

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The value that a function $f(x)$ approaches as $x$ approaches a point $a$ is called the limit. It is formally defined as $\lim_{x\to a}f(x)=L$ if values of $f(x)$ get arbitrarily close to $L$ as $x$ gets arbitrarily close to $a$.

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Derivative

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The derivative measures the instantaneous rate of change of a function. It is defined as $f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$ when this limit exists.

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Integral

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An integral represents the accumulation of quantities and the area under a curve. The indefinite integral is an antiderivative $F(x)$ with $F'(x)=f(x)$, and the definite integral $\int_a^b f(x)\,dx$ gives signed area from $a$ to $b$.

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Chain Rule

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The chain rule gives the derivative of a composite function. If $y=g(f(x))$, then $y'=g'(f(x))\cdot f'(x)$, so you differentiate the outer function at the inner function and multiply by the derivative of the inner function.

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Product Rule

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The product rule provides the derivative of a product of two functions. For $u(x)v(x)$, the derivative is $(uv)'=u'v+uv'$, meaning differentiate one factor and multiply by the other, then add the reverse.

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Quotient Rule

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The quotient rule gives the derivative of a ratio of two functions. For $\frac{u(x)}{v(x)}$, the derivative is $\left(\frac{u}{v}\right)'=\frac{u'v-uv'}{v^2}$, provided $v(x)\neq0$.

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Continuity

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A function is continuous at $a$ if $\lim_{x\to a}f(x)=f(a)$. This requires the limit to exist, the function to be defined at $a$, and the limit to equal the function value.

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Differentiability

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A function is differentiable at a point if its derivative exists there. Differentiability implies continuity, but continuity does not necessarily imply differentiability.

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Mean Value Theorem

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The Mean Value Theorem states that if $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists $c\in(a,b)$ with $f'(c)=\frac{f(b)-f(a)}{b-a}$. It guarantees a point where the instantaneous rate equals the average rate.

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Fundamental Theorem

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The Fundamental Theorem of Calculus links differentiation and integration. Part 1 says $\frac{d}{dx}\left(\int_a^x f(t)\,dt\right)=f(x)$ for continuous $f$, and Part 2 states $\int_a^b f(x)\,dx=F(b)-F(a)$ for any antiderivative $F$.

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L'Hôpital's Rule

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L'Hôpital's Rule helps evaluate limits that yield indeterminate forms $\frac{0}{0}$ or $\frac{\infty}{\infty}$. It says $\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f'(x)}{g'(x)}$ if the latter limit exists.

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Taylor Series

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A Taylor series represents a function as an infinite polynomial centered at a point $a$. It is $f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n$ when the series converges to $f(x)$.

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Critical Point

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A critical point of $f$ is where $f'(x)=0$ or $f'$ is undefined. Critical points are candidates for local maxima, minima, or saddle points and are used in optimization.

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Inflection Point

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An inflection point is where the concavity of a function changes sign. It often occurs where the second derivative $f''(x)$ is zero or undefined, and the sign of $f''$ changes around that point.

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Implicit Differentiation

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Implicit differentiation finds derivatives when $y$ is given implicitly by an equation involving $x$ and $y$. Differentiate both sides with respect to $x$, apply $\frac{d}{dx}y=y'$ for terms with $y$, and solve for $y'$.

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Partial Derivative

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A partial derivative measures the rate of change of a multivariable function with respect to one variable while holding others constant. For $f(x,y)$, the partial $\frac{\partial f}{\partial x}$ treats $y$ as constant and differentiates with respect to $x$.

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Gradient

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The gradient is a vector of all partial derivatives and points in the direction of steepest increase. For $f(x,y)$, $\nabla f=(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y})$, and it is orthogonal to level curves.

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Double Integral

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A double integral $\iint_R f(x,y)\,dA$ computes volume under a surface over a region $R$ in the plane. It can be evaluated as an iterated integral using Fubini's Theorem when $f$ is integrable.

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Improper Integral

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An improper integral has infinite limits or an integrand with unbounded behavior. Convergence is determined by taking limits, e.g., $\int_a^{\infty}f(x)\,dx=\lim_{b\to\infty}\int_a^b f(x)\,dx$, and checking if the limit exists.