Comprehensive Calculus Differentiation Integration and Series Reference Notes

Trigonometric Limit Identities

  • Essential Limits:

    • lim⁡x→0xsin⁡(x)=1\lim_{x \rightarrow 0} \frac{x}{\sin(x)} = 1

    • lim⁡x→0sin⁡(x)x=1\lim_{x \rightarrow 0} \frac{\sin(x)}{x} = 1

    • lim⁡x→01−cos⁡(x)x=0\lim_{x \rightarrow 0} \frac{1-\cos(x)}{x} = 0

    • lim⁡x→0cos⁡(x)−1x=0\lim_{x \rightarrow 0} \frac{\cos(x)-1}{x} = 0

Differentiation Rules: Foundational and Basic Operations

  • Power Rule: ddx[xn]=nxn−1\frac{d}{dx}[x^n] = nx^{n-1}

  • Chain Rule: ddx[f(g(x))]=g′(x)⋅f′(g(x))\frac{d}{dx}[f(g(x))] = g'(x) \cdot f'(g(x))

  • Inverse Rule: ddx[f−1(x)]=1f′(f−1(x))\frac{d}{dx}[f^{-1}(x)] = \frac{1}{f'(f^{-1}(x))}

  • Constant and Constant Multiple Rules:

    • ddx[c]=0\frac{d}{dx}[c] = 0

    • ddx[c⋅f(x)]=c⋅f′(x)\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)

  • Product and Quotient Rules:

    • ddx[f(x)⋅g(x)]=f′(x)g(x)+f(x)g′(x)\frac{d}{dx}[f(x) \cdot g(x)] = f'(x)g(x) + f(x)g'(x)

    • ddx[f(x)g(x)]=f′(x)g(x)−f(x)g′(x)[g(x)]2\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}

  • Euler’s Number and Logarithmic Rules:

    • ddx[ex]=ex\frac{d}{dx}[e^x] = e^x

    • ddx[ln⁡(x)]=1x\frac{d}{dx}[\ln(x)] = \frac{1}{x}

    • ddx[ax]=axln⁡(a)\frac{d}{dx}[a^x] = a^x\ln(a)

    • ddxlog⁡a(x)=1xln⁡(a)\frac{d}{dx}\log_a(x) = \frac{1}{x\ln(a)} where a > 0, a \neq 1

Trigonometric and Inverse Trigonometric Differentiation

  • Primary Functions (Note: Cofunction results in a negative derivative):

    • ddx[sin⁡(x)]=cos⁡(x)\frac{d}{dx}[\sin(x)] = \cos(x)

    • ddx[cos⁡(x)]=−sin⁡(x)\frac{d}{dx}[\cos(x)] = -\sin(x)

    • ddx[tan⁡(x)]=sec⁡2(x)\frac{d}{dx}[\tan(x)] = \sec^2(x)

    • ddx[cot⁡(x)]=−csc⁡2(x)\frac{d}{dx}[\cot(x)] = -\csc^2(x)

    • ddx[sec⁡(x)]=sec⁡(x)tan⁡(x)\frac{d}{dx}[\sec(x)] = \sec(x)\tan(x)

    • ddx[csc⁡(x)]=−csc⁡(x)cot⁡(x)\frac{d}{dx}[\csc(x)] = -\csc(x)\cot(x)

  • Inverse Trigonometric Functions:

    • ddx[sin⁡−1(x)]=11−x2\frac{d}{dx}[\sin^{-1}(x)] = \frac{1}{\sqrt{1-x^2}}

    • ddx[cos⁡−1(x)]=−11−x2\frac{d}{dx}[\cos^{-1}(x)] = -\frac{1}{\sqrt{1-x^2}}

    • ddx[tan⁡−1(x)]=1x2+1\frac{d}{dx}[\tan^{-1}(x)] = \frac{1}{x^2+1}

    • ddx[cot⁡−1(x)]=−1x2+1\frac{d}{dx}[\cot^{-1}(x)] = -\frac{1}{x^2+1}

    • ddx[sec⁡−1(x)]=1∣x∣x2−1\frac{d}{dx}[\sec^{-1}(x)] = \frac{1}{|x|\sqrt{x^2-1}}

    • ddx[csc⁡−1(x)]=−1∣x∣x2−1\frac{d}{dx}[\csc^{-1}(x)] = -\frac{1}{|x|\sqrt{x^2-1}}

Calculus Theorems, Tests, and Fundamental Rules

  • Mean Value Theorem (MVT): On a continuous and differentiable interval (a,b)(a, b), Instantaneous Rate of Change (IROC) must equal Average Rate of Change (AROC) at least once.

  • Intermediate Value Theorem (IVT): On a continuous interval (a,b)(a, b), f(x)f(x) must equal all values between f(a)f(a) and f(b)f(b) at least once.

  • Derivative Tests:

    • 1st Derivative Test identify local extrema and intervals of increasing/decreasing behavior.

    • 2nd Derivative Test determines concavity.

    • Candidates Test identifies global/absolute extrema.

  • L'Hopital's Rule: Used when lim⁡x→af(x)g(x)\lim_{x \rightarrow a} \frac{f(x)}{g(x)} is indeterminate; equals lim⁡x→af′(x)g′(x)\lim_{x \rightarrow a} \frac{f'(x)}{g'(x)}.

  • Limit of a Riemann Sum: lim⁡n→∞∑i=1n(b−an)f(a+b−ani)=lim⁡n→∞∑i=1nΔxf(xi)\lim_{n \rightarrow \infty} \sum_{i=1}^n \left( \frac{b-a}{n} \right) f\left(a + \frac{b-a}{n}i\right) = \lim_{n \rightarrow \infty} \sum_{i=1}^n \Delta x f(x_i).

  • Fundamental Theorem of Calculus (FTC):

    • Standard: ddx∫axf(t)dt=f(x)\frac{d}{dx} \int_{a}^x f(t)dt = f(x)

    • 1st Variation: ddx∫ag(x)f(t)dt=g′(x)⋅f(g(x))\frac{d}{dx} \int_{a}^{g(x)} f(t)dt = g'(x) \cdot f(g(x))

    • 2nd Variation: ddx∫h(x)g(x)f(t)dt=g′(x)⋅f(g(x))−h′(x)⋅f(h(x))\frac{d}{dx} \int_{h(x)}^{g(x)} f(t)dt = g'(x) \cdot f(g(x)) - h'(x) \cdot f(h(x))

Integration and Antiderivative Rules

  • Basic Forms:

    • ∫exdx=ex+C\int e^x dx = e^x + C

    • ∫1xdx=ln⁡∣x∣+C\int \frac{1}{x} dx = \ln|x| + C

    • ∫axdx=1ln⁡(a)ax+C\int a^x dx = \frac{1}{\ln(a)} a^x + C

  • Trigonometry:

    • ∫cos⁡(x)dx=sin⁡(x)+C\int \cos(x) dx = \sin(x) + C

    • ∫sin⁡(x)dx=−cos⁡(x)+C\int \sin(x) dx = -\cos(x) + C

    • ∫sec⁡2(x)dx=tan⁡(x)+C\int \sec^2(x) dx = \tan(x) + C

    • ∫csc⁡2(x)dx=−cot⁡(x)+C\int \csc^2(x) dx = -\cot(x) + C

    • ∫sec⁡(x)tan⁡(x)dx=sec⁡(x)+C\int \sec(x)\tan(x) dx = \sec(x) + C

    • ∫csc⁡(x)cot⁡(x)dx=−csc⁡(x)+C\int \csc(x)\cot(x) dx = -\csc(x) + C

  • Inverse Functions:

    • ∫11−x2dx=sin⁡−1(x)+C\int \frac{1}{\sqrt{1-x^2}} dx = \sin^{-1}(x) + C

    • ∫1x2+1dx=tan⁡−1(x)+C\int \frac{1}{x^2+1} dx = \tan^{-1}(x) + C

    • ∫1∣x∣x2−1dx=sec⁡−1(x)+C\int \frac{1}{|x|\sqrt{x^2-1}} dx = \sec^{-1}(x) + C

Sequences and Series

  • Arithmetic (common difference dd, first term aa):

    • Sequence: tn=a+(n−1)dt_n = a + (n-1)d

    • Series: Sn=n2(2a+(n−1)d)S_n = \frac{n}{2}(2a + (n-1)d)

  • Geometric (common ratio rr, first term aa):

    • Sequence: tn=arn−1t_n = ar^{n-1}

    • Finite Series: Sn=a(1−rn1−r)S_n = a\left(\frac{1-r^n}{1-r}\right)

    • Infinite Series: S∞=a1−rS_{\infty} = \frac{a}{1-r} for |r| < 1

  • Convergence Types:

    • Absolute Convergence: Original series and its absolute value both converge.

    • Conditional Convergence: Original series converges, but its absolute value diverges.

Convergence Tests for Infinite Series

  • Integral Test: For a positive, continuous, decreasing function f(x)f(x) where f(k)=akf(k) = a_k, the series and ∫1∞f(x)dx\int_1^{\infty} f(x)dx share behavior.

  • P-Series Test: ∑k=1∞1kp\sum_{k=1}^{\infty} \frac{1}{k^p} converges if p > 1 and diverges if 0 < p \leq 1.

  • Comparison Test: If 0≤an≤bn0 \leq a_n \leq b_n, then convergence of ∑bk\sum b_k implies convergence of ∑ak\sum a_k, and divergence of ∑ak\sum a_k implies divergence of ∑bk\sum b_k.

  • Ratio Test: For positive series where ρ=lim⁡k→∞ak+1ak\rho = \lim_{k \rightarrow \infty} \frac{a_{k+1}}{a_k}, series converges if \rho < 1, diverges if \rho > 1, and is inconclusive if ρ=1\rho = 1.

  • nth-Term Test: Series diverges if lim⁡k→∞ak≠0\lim_{k \rightarrow \infty} a_k \neq 0.

  • Alternating Series Test: Series converges if terms decrease (ak+1≤aka_{k+1} \leq a_k) and pass the nth-term test (lim⁡k→∞ak=0\lim_{k \rightarrow \infty} a_k = 0).

  • Geometric Series Test: ∑arn\sum ar^n converges if |r| < 1, diverges if ∣r∣≥1|r| \geq 1.

  • Limit Comparison Test: If 0 < c < \infty where c=lim⁡k→∞akbkc = \lim_{k \rightarrow \infty} \frac{a_k}{b_k}, unknown series ∑ak\sum a_k shares behavior with known series ∑bk\sum b_k.

Taylor and Maclaurin Series

  • General Formulas:

    • Taylor Series (about x=ax=a): ∑n=0∞f(n)(a)n!(x−a)n\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n

    • Maclaurin Series (about x=0x=0): ∑n=0∞f(n)(0)n!xn\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n

  • Common Maclaurin Series:

    • sin⁡(x)=∑k=0∞(−1)k(2k+1)!x2k+1\sin(x) = \sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)!}x^{2k+1}

    • cos⁡(x)=∑k=0∞(−1)k(2k)!x2k\cos(x) = \sum_{k=0}^{\infty} \frac{(-1)^k}{(2k)!}x^{2k}

    • ex=∑n=0∞1n!xne^x = \sum_{n=0}^{\infty} \frac{1}{n!}x^n

    • 11−x=∑x=0∞xn\frac{1}{1-x} = \sum_{x=0}^{\infty} x^n

  • Power Series: ∑k=1∞xk=11−x\sum_{k=1}^{\infty} x^k = \frac{1}{1-x} for |x| < 1

Error Bound Calculations

  • Alternating Series Error Bound: If SS is the actual sum and SkS_k is an approximation, the error ∣S−Sk∣≤ak+1|S - S_k| \leq a_{k+1}.

  • Lagrange Error Bound: If Pn(x)P_n(x) is the Taylor approximation of f(x)f(x) about x=ax=a, the error ∣En(x)∣≤M(n+1)!∣x−a∣n+1|E_n(x)| \leq \frac{M}{(n+1)!} |x-a|^{n+1}, where MM is the maximum of ∣f(n+1)(x)∣|f^{(n+1)}(x)| on the interval between xx and aa.