AP Calculus BC - Core Concepts & Definitions

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What is the formal definition of the limit of a function?

Back

The limit of $f(x)$ as $x$ approaches $c$ is $L$ if $f(x)$ can be made arbitrarily close to $L$ by taking $x$ sufficiently close to $c$ (but not equal to $c$). This is the foundation of calculus for defining instantaneous rates of change.

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What does the Intermediate Value Theorem (IVT) state?

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If $f$ is continuous on the closed interval $[a, b]$ and $k$ is any number between $f(a)$ and $f(b)$, then there exists at least one number $c$ in $[a, b]$ such that $f(c) = k$. It guarantees the existence of $x$-values for specific $y$-values.

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Define the derivative of a function at a specific point.

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The derivative of $f$ at $x=a$ is defined as the limit of the difference quotient: $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$, provided the limit exists. Geometrically, this represents the slope of the tangent line to the curve at that point.

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What is the Power Rule for differentiation?

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For any real number $n$, the derivative of $x^n$ with respect to $x$ is $nx^{n-1}$. Formally, if $f(x) = x^n$, then $f'(x) = nx^{n-1}$. This is the most fundamental rule for differentiating polynomial functions.

Front

State the Product Rule for differentiation.

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The derivative of the product of two differentiable functions is the first function times the derivative of the second, plus the second function times the derivative of the first. Formula: $\frac{d}{dx}[f(x)g(x)] = f(x)g'(x) + g(x)f'(x)$.