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Application of Derivatives — Flashcards

Madhya Pradesh Board · Class 12 · Mathematics

60 flashcards for Application of Derivatives (Madhya Pradesh Board Class 12 Mathematics) to test yourself on key terms and facts.

145 questions60 flashcards18 formulas & key relations5 concepts

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An infographic explaining the concept of a derivative as the instantaneous rate of change of one quantity with respect to another, using a simple example like distance vs. time.
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60 Flashcards·
Rate of changeपरिवर्तन की दरसंबंधित दरेंRelated rates
Card 1Rate of change

Find the rate of change of the area of a circle with respect to its radius when r = 5 cm.

Answer

Step 1: Use the area formula A = πr². Step 2: Differentiate with respect to r: dA/dr = 2πr. Step 3: Substitute r = 5 cm: dA/dr = 2π(5) = 10π. Answer: The area changes at the rate of 10π cm² per unit i…

Card 2परिवर्तन की दर

r = 5 cm होने पर एक वृत्त के क्षेत्रफल की उसकी त्रिज्या के सापेक्ष परिवर्तन की दर ज्ञात कीजिए।

Answer

चरण 1: क्षेत्रफल का सूत्र A = πr² का उपयोग कीजिए। चरण 2: r के सापेक्ष अवकलन कीजिए: dA/dr = 2πr. चरण 3: r = 5 cm रखने पर: dA/dr = 2π(5) = 10π. उत्तर: त्रिज्या में 1 इकाई की वृद्धि पर क्षेत्रफल 10π cm² …

Card 3संबंधित दरें

एक घन की भुजा की लंबाई x है। यदि उसका आयतन V = x³ और पृष्ठीय क्षेत्रफल S = 6x² है, तथा dV/dt = 9 cm³/s और x = 10 cm है, तो dS/dt ज्ञात कीजिए।

Answer

चरण 1: V = x³ का t के सापेक्ष अवकलन कीजिए। इसलिए, dV/dt = 3x²(dx/dt). चरण 2: दिया है dV/dt = 9, इसलिए 9 = 3x²(dx/dt). चरण 3: dx/dt के लिए हल कीजिए: dx/dt = 9/(3x²) = 3/x². चरण 4: x = 10 cm पर, dx/dt =…

Card 4Related rates

A cube has edge length x. If its volume is V = x³ and surface area is S = 6x², find dS/dt when dV/dt = 9 cm³/s and x = 10 cm.

Answer

Step 1: Differentiate V = x³ with respect to t. So, dV/dt = 3x²(dx/dt). Step 2: Given dV/dt = 9, so 9 = 3x²(dx/dt). Step 3: Solve for dx/dt: dx/dt = 9/(3x²) = 3/x². Step 4: At x = 10 cm, dx/dt = 3/100…

Card 5Related rates

A stone is dropped into a quiet lake and waves form circles. If the radius increases at 4 cm/s and the radius is 10 cm, find how fast the enclosed area increases.

Answer

Step 1: Use A = πr². Step 2: Differentiate with respect to t. So, dA/dt = 2πr(dr/dt). Step 3: Substitute r = 10 cm and dr/dt = 4 cm/s. Then dA/dt = 2π(10)(4) = 80π. Answer: The area increases at 80π c…

Card 6संबंधित दरें

एक पत्थर को शांत झील में गिराया जाता है और तरंगें वृत्त बनाती हैं। यदि त्रिज्या 4 cm/s की दर से बढ़ रही है और त्रिज्या 10 cm है, तो घिरा हुआ क्षेत्रफल कितनी तेजी से बढ़ रहा है, ज्ञात कीजिए।

Answer

चरण 1: A = πr² का उपयोग कीजिए। चरण 2: t के सापेक्ष अवकलन कीजिए। इसलिए, dA/dt = 2πr(dr/dt). चरण 3: r = 10 cm और dr/dt = 4 cm/s रखिए। तब dA/dt = 2π(10)(4) = 80π. उत्तर: क्षेत्रफल 80π cm²/s की दर से बढ़त…

Card 7संबंधित दरें

एक आयत की लंबाई 3 cm/min की दर से घट रही है और चौड़ाई 2 cm/min की दर से बढ़ रही है। यदि x = 10 cm और y = 6 cm है, तो परिमाप और क्षेत्रफल की परिवर्तन दर ज्ञात कीजिए।

Answer

चरण 1: दिया है dx/dt = -3 cm/min और dy/dt = 2 cm/min. चरण 2: परिमाप P = 2(x + y). अवकलन कीजिए: dP/dt = 2(dx/dt + dy/dt). इसलिए, dP/dt = 2(-3 + 2) = -2 cm/min. चरण 3: क्षेत्रफल A = x·y. अवकलन कीजिए: dA…

Card 8Related rates

The length of a rectangle is decreasing at 3 cm/min and width is increasing at 2 cm/min. If x = 10 cm and y = 6 cm, find the rate of change of perimeter and area.

Answer

Step 1: Given dx/dt = -3 cm/min and dy/dt = 2 cm/min. Step 2: Perimeter P = 2(x + y). Differentiate: dP/dt = 2(dx/dt + dy/dt). So, dP/dt = 2(-3 + 2) = -2 cm/min. Step 3: Area A = x·y. Differentiate: d…

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Frequently Asked Questions

What are the important topics in Application of Derivatives for Madhya Pradesh Board Class 12 Mathematics?
Key topics in Application of Derivatives include Rate of Change of Quantities, Increasing and Decreasing Functions, Local Maxima, Local Minima, and Critical Points, Absolute Maximum and Minimum on a Closed Interval. Study these first, then practise questions on each for the Madhya Pradesh Board Class 12 board exam.
How many flashcards are available for Application of Derivatives?
There are 60 flashcards for Application of Derivatives covering key definitions, facts and ideas. A few sample cards are shown on this page.
How should I revise Application of Derivatives for the Madhya Pradesh Board Class 12 board exam?
Learn the core ideas first, then work through the 145 practice questions on Application of Derivatives. Revise definitions regularly and use flashcards for quick recall before the exam.

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