Gradients — Practice Problems and Worked Solutions

The formulas you need here

18 worked problems

Every step is generated and checked by a computer algebra system.

1. 1.

   Evaluate the gradient at the given point.

   $$f=x^2+y^2,P=\left(1,2\right)$$

   Answer: $\left[\begin{array}{c}2\\ 4\end{array}\right]$

   Step-by-step solution

   1. $$\nabla f=\left[\begin{array}{c}2x\\ 2y\end{array}\right]$$
   2. $$\nabla f\left(1,2\right)=\left[\begin{array}{c}2\\ 4\end{array}\right]$$
2. 2.

   Evaluate the gradient at the given point.

   $$f=xy+y^2,P=\left(2,-1\right)$$

   Answer: $\left[\begin{array}{c}-1\\ 0\end{array}\right]$

   Step-by-step solution

   1. $$\nabla f=\left[\begin{array}{c}y\\ x+2y\end{array}\right]$$
   2. $$\nabla f\left(2,-1\right)=\left[\begin{array}{c}-1\\ 0\end{array}\right]$$
3. 3.

   Evaluate the gradient at the given point.

   $$f=x^3-3xy,P=\left(1,1\right)$$

   Answer: $\left[\begin{array}{c}0\\ -3\end{array}\right]$

   Step-by-step solution

   1. $$\nabla f=\left[\begin{array}{c}3x^2-3y\\ -3x\end{array}\right]$$
   2. $$\nabla f\left(1,1\right)=\left[\begin{array}{c}0\\ -3\end{array}\right]$$
4. 4.

   Evaluate the gradient at the given point.

   $$f=xy+e^x,P=\left(0,2\right)$$

   Answer: $\left[\begin{array}{c}3\\ 0\end{array}\right]$

   Step-by-step solution

   1. $$\nabla f=\left[\begin{array}{c}y+e^x\\ x\end{array}\right]$$
   2. $$\nabla f\left(0,2\right)=\left[\begin{array}{c}3\\ 0\end{array}\right]$$
5. 5.

   Evaluate the gradient at the given point.

   $$f=x^{2}y+y^3,P=\left(-1,1\right)$$

   Answer: $\left[\begin{array}{c}-2\\ 4\end{array}\right]$

   Step-by-step solution

   1. $$\nabla f=\left[\begin{array}{c}2xy\\ x^2+3y^2\end{array}\right]$$
   2. $$\nabla f\left(-1,1\right)=\left[\begin{array}{c}-2\\ 4\end{array}\right]$$
6. 6.

   Evaluate the gradient at the given point.

   $$f=x+\sin \left(xy\right),P=\left(0,3\right)$$

   Answer: $\left[\begin{array}{c}4\\ 0\end{array}\right]$

   Step-by-step solution

   1. $$\nabla f=\left[\begin{array}{c}y\cos \left(xy\right)+1\\ x\cos \left(xy\right)\end{array}\right]$$
   2. $$\nabla f\left(0,3\right)=\left[\begin{array}{c}4\\ 0\end{array}\right]$$
7. 7.

   Evaluate the gradient at the given point.

   $$f=x^2+3y,P=\left(2,1\right)$$

   Answer: $\left[\begin{array}{c}4\\ 3\end{array}\right]$

   Step-by-step solution

   1. $$\nabla f=\left[\begin{array}{c}2x\\ 3\end{array}\right]$$
   2. $$\nabla f\left(2,1\right)=\left[\begin{array}{c}4\\ 3\end{array}\right]$$
8. 8.

   Evaluate the gradient at the given point.

   $$f=xy,P=\left(3,2\right)$$

   Answer: $\left[\begin{array}{c}2\\ 3\end{array}\right]$

   Step-by-step solution

   1. $$\nabla f=\left[\begin{array}{c}y\\ x\end{array}\right]$$
   2. $$\nabla f\left(3,2\right)=\left[\begin{array}{c}2\\ 3\end{array}\right]$$
9. 9.

   Evaluate the gradient at the given point.

   $$f=x^3+y^2,P=\left(1,-2\right)$$

   Answer: $\left[\begin{array}{c}3\\ -4\end{array}\right]$

   Step-by-step solution

   1. $$\nabla f=\left[\begin{array}{c}3x^2\\ 2y\end{array}\right]$$
   2. $$\nabla f\left(1,-2\right)=\left[\begin{array}{c}3\\ -4\end{array}\right]$$
10. 10.

    Evaluate the gradient at the given point.

    $$f=2x^2+y^2,P=\left(1,3\right)$$

    Answer: $\left[\begin{array}{c}4\\ 6\end{array}\right]$

    Step-by-step solution

    1. $$\nabla f=\left[\begin{array}{c}4x\\ 2y\end{array}\right]$$
    2. $$\nabla f\left(1,3\right)=\left[\begin{array}{c}4\\ 6\end{array}\right]$$
11. 11.

    Evaluate the gradient at the given point.

    $$f=x{y}^2,P=\left(2,1\right)$$

    Answer: $\left[\begin{array}{c}1\\ 4\end{array}\right]$

    Step-by-step solution

    1. $$\nabla f=\left[\begin{array}{c}{y}^2\\ 2xy\end{array}\right]$$
    2. $$\nabla f\left(2,1\right)=\left[\begin{array}{c}1\\ 4\end{array}\right]$$
12. 12.

    Evaluate the gradient at the given point.

    $$f=x^{2}y,P=\left(-1,2\right)$$

    Answer: $\left[\begin{array}{c}-4\\ 1\end{array}\right]$

    Step-by-step solution

    1. $$\nabla f=\left[\begin{array}{c}2xy\\ x^2\end{array}\right]$$
    2. $$\nabla f\left(-1,2\right)=\left[\begin{array}{c}-4\\ 1\end{array}\right]$$
13. 13.

    Evaluate the gradient at the given point.

    $$f=y+e^x,P=\left(0,5\right)$$

    Answer: $\left[\begin{array}{c}1\\ 1\end{array}\right]$

    Step-by-step solution

    1. $$\nabla f=\left[\begin{array}{c}{e}^x\\ 1\end{array}\right]$$
    2. $$\nabla f\left(0,5\right)=\left[\begin{array}{c}1\\ 1\end{array}\right]$$
14. 14.

    Evaluate the gradient at the given point.

    $$f=x^2-y^2,P=\left(3,1\right)$$

    Answer: $\left[\begin{array}{c}6\\ -2\end{array}\right]$

    Step-by-step solution

    1. $$\nabla f=\left[\begin{array}{c}2x\\ -2y\end{array}\right]$$
    2. $$\nabla f\left(3,1\right)=\left[\begin{array}{c}6\\ -2\end{array}\right]$$
15. 15.

    Evaluate the gradient at the given point.

    $$f=xy+x,P=\left(1,4\right)$$

    Answer: $\left[\begin{array}{c}5\\ 1\end{array}\right]$

    Step-by-step solution

    1. $$\nabla f=\left[\begin{array}{c}y+1\\ x\end{array}\right]$$
    2. $$\nabla f\left(1,4\right)=\left[\begin{array}{c}5\\ 1\end{array}\right]$$
16. 16.

    Evaluate the gradient at the given point.

    $$f=x+y^3,P=\left(2,1\right)$$

    Answer: $\left[\begin{array}{c}1\\ 3\end{array}\right]$

    Step-by-step solution

    1. $$\nabla f=\left[\begin{array}{c}1\\ 3y^2\end{array}\right]$$
    2. $$\nabla f\left(2,1\right)=\left[\begin{array}{c}1\\ 3\end{array}\right]$$
17. 17.

    Evaluate the gradient at the given point.

    $$f=y^2+\cos \left(x\right),P=\left(0,2\right)$$

    Answer: $\left[\begin{array}{c}0\\ 4\end{array}\right]$

    Step-by-step solution

    1. $$\nabla f=\left[\begin{array}{c}-\sin \left(x\right)\\ 2y\end{array}\right]$$
    2. $$\nabla f\left(0,2\right)=\left[\begin{array}{c}0\\ 4\end{array}\right]$$
18. 18.

    Evaluate the gradient at the given point.

    $$f=x^2{y}^2,P=\left(1,1\right)$$

    Answer: $\left[\begin{array}{c}2\\ 2\end{array}\right]$

    Step-by-step solution

    1. $$\nabla f=\left[\begin{array}{c}2x{y}^2\\ 2x^{2}y\end{array}\right]$$
    2. $$\nabla f\left(1,1\right)=\left[\begin{array}{c}2\\ 2\end{array}\right]$$

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• Partial Derivatives
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• Double Integrals
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• Divergence & Curl