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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 <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><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></math>, find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>&#x2202;</mo><mi>f</mi></mrow><mrow><mo>&#x2202;</mo><mi>x</mi></mrow></mfrac></math>.


Answer: <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>&#x2202;</mo><mi>f</mi></mrow><mrow><mo>&#x2202;</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>&#x2202;</mo><mi>f</mi></mrow><mrow><mo>&#x2202;</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mo>&#x2202;</mo><mrow><mo>&#x2202;</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 <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><mn>2</mn><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi></math>, find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>&#x2202;</mo><mi>f</mi></mrow><mrow><mo>&#x2202;</mo><mi>y</mi></mrow></mfrac></math>.


Answer: <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>&#x2202;</mo><mi>f</mi></mrow><mrow><mo>&#x2202;</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>&#x2202;</mo><mi>f</mi></mrow><mrow><mo>&#x2202;</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>&#x2202;</mo><mi>f</mi></mrow><mrow><mo>&#x2202;</mo><mi>y</mi></mrow></mfrac></math>


Answer: <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>&#x2202;</mo><mi>f</mi></mrow><mrow><mo>&#x2202;</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>&#x2202;</mo><mrow><mo>&#x2202;</mo><mi>y</mi></mrow></mfrac><mfenced><mrow><mi>s</mi><mi>e</mi><mi>c</mi><mfenced><mi>y</mi></mfenced><mo>&#xB7;</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>&#x2202;</mo><mi>f</mi></mrow><mrow><mo>&#x2202;</mo><mi>y</mi></mrow></mfrac><mo>=</mo><mfrac><mo>&#x2202;</mo><mrow><mo>&#x2202;</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: <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><mn>2</mn><msup><mi>e</mi><mrow><mi>x</mi><mi>y</mi></mrow></msup></math> Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>&#x2202;</mo><mn>2</mn></msup><mi>f</mi></mrow><mrow><mo>&#x2202;</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>&#x2202;</mo><mn>2</mn></msup><mi>f</mi></mrow><mrow><mo>&#x2202;</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>&#xB7;</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: <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><mo>-</mo><mn>6</mn><msup><mi>e</mi><mi>y</mi></msup><mi>x</mi></math> Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>&#x2202;</mo><mn>2</mn></msup><mi>f</mi></mrow><mrow><mo>&#x2202;</mo><mi>y</mi><mo>&#x2202;</mo><mi>x</mi></mrow></mfrac></math>.

Answer: <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>&#x2202;</mo><mn>2</mn></msup><mi>f</mi></mrow><mrow><mo>&#x2202;</mo><mi>y</mi><mo>&#x2202;</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: <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="left"><mtr><mtd><mi>f</mi><mfenced><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi></mrow></mfenced><mo>=</mo><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mi>z</mi></mtd></mtr><mtr><mtd><mi>g</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mfenced><mrow><msup><mi>e</mi><mi>t</mi></msup><mo>,</mo><mi>sin</mi><mfenced><mi>t</mi></mfenced><mo>,</mo><mi>t</mi></mrow></mfenced></mtd></mtr></mtable></math>

Find h’(t) if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mi>f</mi><mfenced><mrow><mi>g</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced></math>

Answer: <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>h</mi><mo>'</mo></msup><mfenced><mi>t</mi></mfenced><mo>=</mo><msup><mi>e</mi><mi>t</mi></msup><mo>+</mo><mn>2</mn><mi>cos</mi><mfenced><mi>t</mi></mfenced><mo>&#xB7;</mo><mi>t</mi><mo>+</mo><mn>2</mn><mi>sin</mi><mfenced><mi>t</mi></mfenced></math>

Explanation: Write an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mi>f</mi><mfenced><mrow><mi>g</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced></math>

 

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Question 7: Given

<math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="left"><mtr><mtd><mi>g</mi><mfenced><mn>1</mn></mfenced><mo>=</mo><mfenced><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>6</mn></mrow></mfenced></mtd></mtr><mtr><mtd><msup><mi>g</mi><mo>'</mo></msup><mfenced><mn>1</mn></mfenced><mo>=</mo><mfenced><mrow><mn>0</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>2</mn></mrow></mfenced></mtd></mtr></mtable></math>

And, 

<math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="left"><mtr><mtd><mi>h</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mi>f</mi><mfenced><mrow><mi>g</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced></mtd></mtr><mtr><mtd><mo>&#x2207;</mo><mi>f</mi><mfenced><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>6</mn></mrow></mfenced><mo>=</mo><mfenced><mrow><mn>4</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>1</mn></mrow></mfenced></mtd></mtr></mtable></math>

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: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mfenced><mn>1</mn></mfenced><mo>=</mo><mn>11</mn></math>

Explanation: Apply the formula: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mfenced><mn>1</mn></mfenced><mo>=</mo><mo>&#x2207;</mo><mi>f</mi><mo>(</mo><mi>g</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo><mo>&#x22C5;</mo><mi>g</mi><mo>'</mo><mo>(</mo><mn>1</mn><mo>)</mo></math>

 

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Question 8: Given

<math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="left"><mtr><mtd><mi>g</mi><mfenced><mn>2</mn></mfenced><mo>=</mo><mfenced><mrow><mo>-</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></mfenced></mtd></mtr><mtr><mtd><msup><mi>g</mi><mo>'</mo></msup><mfenced><mn>2</mn></mfenced><mo>=</mo><mfenced><mrow><mn>5</mn><mo>,</mo><mn>1</mn></mrow></mfenced></mtd></mtr></mtable></math>

And, 

<math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="left"><mtr><mtd><mi>h</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mi>f</mi><mfenced><mrow><mi>g</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced></mtd></mtr><mtr><mtd><mo>&#x2207;</mo><mi>f</mi><mfenced><mrow><mo>-</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></mfenced><mo>=</mo><mfenced><mrow><mn>3</mn><mo>,</mo><mn>5</mn></mrow></mfenced></mtd></mtr></mtable></math>

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: <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>h</mi><mo>'</mo></msup><mfenced><mn>2</mn></mfenced><mo>=</mo><mn>20</mn></math>

Explanation: Apply the formula: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mfenced><mn>2</mn></mfenced><mo>=</mo><mo>&#x2207;</mo><mi>f</mi><mo>(</mo><mi>g</mi><mo>(</mo><mn>2</mn><mo>)</mo><mo>)</mo><mo>&#x22C5;</mo><mi>g</mi><mo>'</mo><mo>(</mo><mn>2</mn><mo>)</mo></math>

 

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Question 9: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo><mo>=</mo><mi>x</mi><mi>y</mi><mo>&#x2212;</mo><mi>z</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mo>(</mo><mn>3</mn><mo>,</mo><mo>-</mo><mi>t</mi><mo>,</mo><mn>2</mn><mi>t</mi><mo>)</mo></math>. Find h’(2) if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mi>f</mi><mfenced><mrow><mi>g</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced></math>.


Answer: <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>h</mi><mo>'</mo></msup><mfenced><mn>2</mn></mfenced><mo>=</mo><mo>-</mo><mn>5</mn></math>

Explanation: Let us find<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x2207;</mo><mi>f</mi></math>.

<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x2207;</mo><mi>f</mi><mo>=</mo><mfenced><mrow><mfrac><mrow><mo>&#x2202;</mo><mi>f</mi></mrow><mrow><mo>&#x2202;</mo><mi>x</mi></mrow></mfrac><mover><mi>i</mi><mo>^</mo></mover><mo>+</mo><mfrac><mrow><mo>&#x2202;</mo><mi>f</mi></mrow><mrow><mo>&#x2202;</mo><mi>y</mi></mrow></mfrac><mover><mi>j</mi><mo>^</mo></mover><mo>+</mo><mfrac><mrow><mo>&#x2202;</mo><mi>f</mi></mrow><mrow><mo>&#x2202;</mo><mi>z</mi></mrow></mfrac><mover><mi>k</mi><mo>^</mo></mover></mrow></mfenced></math>

<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x2207;</mo><mi>f</mi><mo>=</mo><mfenced><mrow><mi>y</mi><mo>,</mo><mi>x</mi><mo>,</mo><mo>-</mo><mn>1</mn></mrow></mfenced></math>

 

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Question 10: Given,

 

<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></mfenced><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mi>sin</mi><mfenced><mi>u</mi></mfenced></mtd></mtr><mtr><mtd><mi>cos</mi><mfenced><mi>v</mi></mfenced></mtd></mtr><mtr><mtd><mi>sin</mi><mfenced><mi>u</mi></mfenced><mo>&#xB7;</mo><mi>cos</mi><mfenced><mi>v</mi></mfenced></mtd></mtr></mtable></mfenced></math>


Answer: <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><mi>cos</mi><mfenced><mi>u</mi></mfenced></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mi>sin</mi><mfenced><mi>v</mi></mfenced></mtd></mtr><mtr><mtd><mi>cos</mi><mfenced><mi>u</mi></mfenced><mo>&#xB7;</mo><mi>cos</mi><mfenced><mi>v</mi></mfenced></mtd><mtd><mi>sin</mi><mfenced><mi>u</mi></mfenced><mo>&#xB7;</mo><mo>-</mo><mi>sin</mi><mfenced><mi>v</mi></mfenced></mtd></mtr></mtable></mfenced></math>


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>&#x2202;</mo><msub><mi>f</mi><mn>0</mn></msub></mrow><mrow><mo>&#x2202;</mo><mi>u</mi></mrow></mfrac></mtd><mtd><mfrac><mrow><mo>&#x2202;</mo><msub><mi>f</mi><mn>0</mn></msub></mrow><mrow><mo>&#x2202;</mo><mi>v</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mfrac><mrow><mo>&#x2202;</mo><msub><mi>f</mi><mn>1</mn></msub></mrow><mrow><mo>&#x2202;</mo><mi>u</mi></mrow></mfrac></mtd><mtd><mfrac><mrow><mo>&#x2202;</mo><msub><mi>f</mi><mn>1</mn></msub></mrow><mrow><mo>&#x2202;</mo><mi>v</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mfrac><mrow><mo>&#x2202;</mo><msub><mi>f</mi><mn>2</mn></msub></mrow><mrow><mo>&#x2202;</mo><mi>u</mi></mrow></mfrac></mtd><mtd><mfrac><mrow><mo>&#x2202;</mo><msub><mi>f</mi><mn>2</mn></msub></mrow><mrow><mo>&#x2202;</mo><mi>v</mi></mrow></mfrac></mtd></mtr></mtable></mfenced></math>

 

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