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एक फ़ंक्शन के निरंतर होने की परिभाषा क्या है?
एक फ़ंक्शन निरंतर है यदि उसके मूल्य और उस स्थान पर सीमा एक समान हैं जब इनपुट उस मान के निकट पहुँचता है।
पॉलीनोमियल फ़ंक्शन के व्युत्क्रम को निकालने का कौन सा नियम उपयोग किया जाता है?
पॉलीनोमियल फ़ंक्शन के व्युत्क्रम को निकालने के लिए पावर नियम का उपयोग किया जाता है।
ट्रिगोनोमेट्रिक फ़ंक्शन के व्युत्क्रम निकालने में कौन सी पहचानें सहायक होती हैं?
साइन, कोसाइन, टैंजेंट, कोटैंजेंट, सेकेंट और कोसेकेंट फ़ंक्शन के व्युत्क्रम ज्ञात पहचानें का उपयोग करके निकाले जाते हैं।
हाइपरबोलिक फ़ंक्शन का उपयोग कहां होता है?
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अविरति (discontinuity) के कौन से प्रकार होते हैं?
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कलनासी में किस तरह के कार्य और परिवर्तन का अध्ययन किया जाता है?
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उपग्रही $x$ पर किसी कार्य का व्युत्पत्ति क्या दर्शाती है?
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व्युत्पत्ति के लिए कौनसे महत्वपूर्ण नियम हैं?
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इंटीग्रल का निश्चित इंटीग्रल किसे दर्शाता है?
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कलनासी के मूलभूत प्रमेय का क्या महत्व है?
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अखंड कार्यों में ऊर्ध्वाधर सीमाओं के क्रिय्रान्वयन का क्या मतलब है?
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सर्वोच्च और न्यूनतम मान खोजने के लिए व्युत्पत्ति का उपयोग कैसे किया जाता है?
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सीमा का क्या महत्व है कलनासी में?
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Study Notes
- Calculus is a branch of mathematics that deals with the study of change.
- It is fundamentally concerned with continuous functions and their rates of change.
- Two major branches are differential calculus (concerned with rates of change) and integral calculus (concerned with accumulation of quantities).
Differential Calculus
- Derivatives: The derivative of a function at a point represents the instantaneous rate of change of the function at that point. Geometrically, it corresponds to the slope of the tangent line to the graph at that point.
- Rules for Differentiation: There are various rules for calculating derivatives, including the power rule, product rule, quotient rule, chain rule, and the rules for differentiating trigonometric, exponential, and logarithmic functions. Each rule allows for efficient derivation of complex functions.
- Applications of Derivatives: These include finding maximum and minimum values of functions, determining concavity and inflection points, and solving optimization problems.
Integral Calculus
- Integrals: The definite integral of a function over an interval represents the area under the curve of the function. The function is the integrand, the interval is the region of integration and the result is the area.
- Techniques of Integration: Various techniques exist for evaluating integrals, including substitution (u-substitution), integration by parts, and trigonometric integrals. Specific techniques are required for different types of integrals to facilitate calculations.
- Applications of Integrals: Applying definite integrals allows for finding areas, volumes of solids of revolution, arc length, work done by a force, and other useful applications.
Fundamental Theorem of Calculus
- This fundamental theorem establishes a connection between differentiation and integration.
- Part 1 states that the derivative of a definite integral, with respect to the upper limit, is the integrand evaluated at that upper limit. This relates to the concept of instantaneous rate of change.
- Part 2 demonstrates how to evaluate definite integrals by using antiderivatives in a concise way. It is a powerful tool that allows for the evaluation of complicated integrals without iterative processes.
Limits
- Limits are crucial to the understanding of both differential and integral calculus.
- They describe the behavior of a function as the input approaches a specific value, or approaches positive or negative infinity.
- Calculating limits rigorously often involves algebraic manipulations or various limit theorems.
Continuity
- A function is continuous at a point if its value at that point equals the limit of the function at that point as the input approaches that value.
- Continuity is essential for many theorems and procedures in calculus. Failure in continuity may lead to various problems in the application.
- Discontinuities come in various forms, such as jump discontinuities, removable discontinuities, and infinite discontinuities.
Common Functions and Their Derivatives
- Polynomials: The derivative of a polynomial function can be found using the power rule. The power rule works consistently for various orders of polynomials.
- Trigonometric functions: The derivatives of sine, cosine, tangent, cotangent, secant, and cosecant functions can be derived using known identities. These are fundamental trigonometric relations.
- Exponential and Logarithmic Functions: The derivatives of exponential and logarithmic functions are essential tools for modelling phenomena involving growth and decay. Common applications range from population dynamics to radioactive decay.
- Hyperbolic Functions: The hyperbolic functions and their derivatives often emerge in more complex problems including special functions.
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Description
इस क्विज में आप डेरिवेटिव, विभिन्न डिफरेंशियेशन नियम और उनके अनुप्रयोगों के बारे में सीखेंगे। यह गणित की एक महत्वपूर्ण शाखा है जो परिवर्तन के अध्ययन से संबंधित है। अधिकतम और न्यूनतम मानों को खोजने के लिए डेरिवेटिव का उपयोग कैसे किया जाता है, यह जानें।
