Multivariable Calculus:

In multivariable calculus, we work on functions of more than one variable. It is an advanced version of single variable calculus. Here, we calculate the partial derivative, which is almost similar to derivatives of a single variable. Integrals of functions having more than one variable are also a part of multivariable calculus.

Multivariable Calculus Sample Questions:

Question 1: For , find $<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>∂</mo><mi>f</mi></mrow><mrow><mo>∂</mo><mi>x</mi></mrow></mfrac></math>$ .

Answer: $<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>∂</mo><mi>f</mi></mrow><mrow><mo>∂</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></math>$

Explanation: $<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>∂</mo><mi>f</mi></mrow><mrow><mo>∂</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>x</mi></mrow></mfrac><mfenced><mrow><mn>2</mn><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi></mrow></mfenced></math>$

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Question 2: For , find $<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>∂</mo><mi>f</mi></mrow><mrow><mo>∂</mo><mi>y</mi></mrow></mfrac></math>$ .

Answer: $<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>∂</mo><mi>f</mi></mrow><mrow><mo>∂</mo><mi>y</mi></mrow></mfrac><mo>=</mo><mn>0</mn></math>$

Explanation: We want to find $<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>∂</mo><mi>f</mi></mrow><mrow><mo>∂</mo><mi>y</mi></mrow></mfrac></math>$ , so we are going to consider other variables as a constant.

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Question 3: $<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mn>2</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><msup><mi>e</mi><mi>x</mi></msup></mrow><mrow><mi>cos</mi><mfenced><mi>y</mi></mfenced></mrow></mfrac></math>$ Find $<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>∂</mo><mi>f</mi></mrow><mrow><mo>∂</mo><mi>y</mi></mrow></mfrac></math>$

Answer: $<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>∂</mo><mi>f</mi></mrow><mrow><mo>∂</mo><mi>y</mi></mrow></mfrac><mo>=</mo><mfenced><mrow><mn>2</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><msup><mi>e</mi><mi>x</mi></msup></mrow></mfenced><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>y</mi></mrow></mfrac><mfenced><mrow><mi>s</mi><mi>e</mi><mi>c</mi><mfenced><mi>y</mi></mfenced><mo>·</mo><mi>tan</mi><mfenced><mi>y</mi></mfenced></mrow></mfenced></math>$

Explanation: $<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>∂</mo><mi>f</mi></mrow><mrow><mo>∂</mo><mi>y</mi></mrow></mfrac><mo>=</mo><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>y</mi></mrow></mfrac><mfenced><mfrac><mrow><mn>2</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><msup><mi>e</mi><mi>x</mi></msup></mrow><mrow><mi>cos</mi><mfenced><mi>y</mi></mfenced></mrow></mfrac></mfenced></math>$

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Question 4: Find $<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>∂</mo><mn>2</mn></msup><mi>f</mi></mrow><mrow><mo>∂</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></math>$

Answer: $<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>∂</mo><mn>2</mn></msup><mi>f</mi></mrow><mrow><mo>∂</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>2</mn><msup><mi>y</mi><mn>2</mn></msup><mo>·</mo><msup><mi>e</mi><mrow><mi>x</mi><mi>y</mi></mrow></msup></math>$

Explanation: We need to find second-order partial derivatives. Let us apply the chain rule.

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Question 5: Find $<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>∂</mo><mn>2</mn></msup><mi>f</mi></mrow><mrow><mo>∂</mo><mi>y</mi><mo>∂</mo><mi>x</mi></mrow></mfrac></math>$ .

Answer: $<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>∂</mo><mn>2</mn></msup><mi>f</mi></mrow><mrow><mo>∂</mo><mi>y</mi><mo>∂</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mn>6</mn><mfenced><msup><mi>e</mi><mi>y</mi></msup></mfenced></math>$

Explanation: First, we will take the partial derivative with respect to x.

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Question 6:

Find h’(t) if

Answer:

Explanation: Write an expression for

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Question 7: Given

And,

Find $<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>d</mi><mi>h</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></math>$ at t=1.

Answer:

Explanation: Apply the formula:

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Question 8: Given

And,

Find $<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>d</mi><mi>h</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></math>$ at t=2.

Answer:

Explanation: Apply the formula:

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Question 9: , . Find h’(2) if .

Answer:

Explanation: Let us find.

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∇</mo><mi>f</mi><mo>=</mo><mfenced><mrow><mfrac><mrow><mo>∂</mo><mi>f</mi></mrow><mrow><mo>∂</mo><mi>x</mi></mrow></mfrac><mover><mi>i</mi><mo>^</mo></mover><mo>+</mo><mfrac><mrow><mo>∂</mo><mi>f</mi></mrow><mrow><mo>∂</mo><mi>y</mi></mrow></mfrac><mover><mi>j</mi><mo>^</mo></mover><mo>+</mo><mfrac><mrow><mo>∂</mo><mi>f</mi></mrow><mrow><mo>∂</mo><mi>z</mi></mrow></mfrac><mover><mi>k</mi><mo>^</mo></mover></mrow></mfenced></math>$

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Question 10: Given,

Answer:

Explanation: We know the Jacobian:

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>J</mi><mfenced><mi>f</mi></mfenced><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mfrac><mrow><mo>∂</mo><msub><mi>f</mi><mn>0</mn></msub></mrow><mrow><mo>∂</mo><mi>u</mi></mrow></mfrac></mtd><mtd><mfrac><mrow><mo>∂</mo><msub><mi>f</mi><mn>0</mn></msub></mrow><mrow><mo>∂</mo><mi>v</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mfrac><mrow><mo>∂</mo><msub><mi>f</mi><mn>1</mn></msub></mrow><mrow><mo>∂</mo><mi>u</mi></mrow></mfrac></mtd><mtd><mfrac><mrow><mo>∂</mo><msub><mi>f</mi><mn>1</mn></msub></mrow><mrow><mo>∂</mo><mi>v</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mfrac><mrow><mo>∂</mo><msub><mi>f</mi><mn>2</mn></msub></mrow><mrow><mo>∂</mo><mi>u</mi></mrow></mfrac></mtd><mtd><mfrac><mrow><mo>∂</mo><msub><mi>f</mi><mn>2</mn></msub></mrow><mrow><mo>∂</mo><mi>v</mi></mrow></mfrac></mtd></mtr></mtable></mfenced></math>$

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