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Calculus: Limits and Derivatives of Trigonometric Functions

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Termini in questo insieme (20)

  • What is the limit of \(\frac{\sin x}{x}\) as \(x \to 0\)?
    The limit is \(1\).
  • What is the limit of \(\frac{\cos x - 1}{x}\) as \(x \to 0\)?
    The limit is \(0\).
  • State the limit definition of the derivative for \(\sin x\).
    The derivative is defined as \(\lim_{h \to 0} \frac{\sin(x+h) - \sin x}{h}\).
  • What is the derivative of \(\sin x\)?
    The derivative is \(\cos x\).
  • What is the derivative of \(\cos x\)?
    The derivative is \(-\sin x\).
  • State the angle addition formula for sine.
    \(\sin(A+B) = \sin A \cos B + \cos A \sin B\).
  • State the angle addition formula for cosine.
    \(\cos(A+B) = \cos A \cos B - \sin A \sin B\).
  • What is the derivative of \(\tan x\)?
    The derivative is \(\sec^2 x\).
  • What is the derivative of \(\sec x\)?
    The derivative is \(\sec x \tan x\).
  • What is the derivative of \(\cot x\)?
    The derivative is \(-\csc^2 x\).
  • What is the derivative of \(\csc x\)?
    The derivative is \(-\csc x \cot x\).
  • How do you apply the quotient rule to find the derivative of \(\tan x = \frac{\sin x}{\cos x}\)?
    Use \(\frac{d}{dx} \left( \frac{N}{D} \right) = \frac{N' D - N D'}{D^2}\) with \(N=\sin x\) and \(D=\cos x\).
  • What is the limit of \(\frac{\sin 3x}{3x}\) as \(x \to 0\)?
    The limit is \(1\).
  • What is the limit of \(\frac{\sin 2x}{x}\) as \(x \to 0\)?
    The limit is \(2\).
  • What is the derivative of \(\sin(At+B)\) using the chain rule?
    The derivative is \(A \cos(At+B)\).
  • What is the derivative of \(\cos(At+B)\) using the chain rule?
    The derivative is \(-A \sin(At+B)\).
  • What is the derivative of \(\sec^2 x\)?
    The derivative is \(2 \sec^2 x \tan x\) by chain rule.
  • What is the derivative of \(\log(2x)\) at \(x=0\)?
    The derivative is undefined at \(x=0\) because \(\log(0)\) is undefined.
  • What is the derivative of \(\tan x\) using the quotient rule?
    Using quotient rule on \(\frac{\sin x}{\cos x}\), derivative is \(\sec^2 x\).
  • What is the derivative of \(\sin x \cos x\) using the product rule?
    The derivative is \(\cos^2 x - \sin^2 x\).