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Calculus 1

The document covers fundamental concepts in calculus, including limits, continuity, derivative rules, common derivatives, and integration basics. It explains the conditions for continuity, various derivative rules such as power, product, quotient, and chain rules, as well as common derivatives of trigonometric and exponential functions. Additionally, it introduces integration as the reverse of differentiation and outlines the power rule for integrals and the Fundamental Theorem of Calculus.

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jasleen2325
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221 views2 pages

Calculus 1

The document covers fundamental concepts in calculus, including limits, continuity, derivative rules, common derivatives, and integration basics. It explains the conditions for continuity, various derivative rules such as power, product, quotient, and chain rules, as well as common derivatives of trigonometric and exponential functions. Additionally, it introduces integration as the reverse of differentiation and outlines the power rule for integrals and the Fundamental Theorem of Calculus.

Uploaded by

jasleen2325
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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🔹 1.

Limits & Continuity


●​ A limit tells you what value a function approaches as the input gets close to something.​

●​ Notation:​
lim⁡x→af(x)=L\lim_{x \to a} f(x) = L
●​ If a function has a hole, jump, or asymptote, it may not be continuous at that point.​

●​ Continuity Rule:​
A function is continuous at x=ax = a if:​

1.​ f(a)f(a) is defined​

2.​ lim⁡x→af(x)\lim_{x \to a} f(x) exists​

3.​ lim⁡x→af(x)=f(a)\lim_{x \to a} f(x) = f(a)​

🔹 2. Derivative Rules
●​ Power Rule:​
ddx[xn]=nxn−1\frac{d}{dx} [x^n] = nx^{n-1}
●​ Product Rule:​
(fg)′=f′g+fg′(fg)' = f'g + fg'
●​ Quotient Rule:​
(fg)′=f′g−fg′g2\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}
●​ Chain Rule:​
ddxf(g(x))=f′(g(x))⋅g′(x)\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)

🔹 3. Common Derivatives
●​ 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⁡2x\frac{d}{dx}[\tan x] = \sec^2 x​

●​ ddx[ex]=ex\frac{d}{dx}[e^x] = e^x​
●​ ddx[ln⁡x]=1x\frac{d}{dx}[\ln x] = \frac{1}{x}​

🔹 4. Integration Basics
●​ Reverse of derivative.​

●​ Power Rule for Integrals:​


∫xndx=xn+1n+1+C(n≠−1)\int x^n dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)
●​ Definite Integrals give area under the curve from aa to bb:​
∫abf(x)dx\int_a^b f(x) dx
●​ Fundamental Theorem of Calculus:​
ddx∫axf(t)dt=f(x)\frac{d}{dx} \int_a^x f(t) dt = f(x)

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