4.2.E: Exercises
- Page ID
- 157595
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For problems 1 through 16, find \(f_x\) and \(f_y\) for the function given
| 1. \(f(x,y) = x^2-5y^2\) |
| 2. \(f(x,y) = \frac{x^2-5y^2}{x+4}\) |
| 3. \(f(x,y) = e^{x+6y}\) |
| 4. \(f(x,y) = (x^2-5y^2) e^x\) |
| 5. \(f(x,y) = (x^2 - 5y^2) \left(\frac{1}{3y}+4\right)\) |
| 6. \(f(x,y) = x\) |
| 7. \(f(x,y) = 6\) |
| 8. \(f(x,y) = \ln (xy+2x-6y) \) |
| 9. \(f(x,y) = \frac{x^2-5y^2}{y^4-5x^4}\) |
| 10. \(f(x,y) = e^{\sqrt{x-4y}} (x-4y)\) |
| 11. \(f(x,y) = y^5 e^x\) |
| 12. \(f(x,y) = \frac{1}{16xy}\) |
| 13. \(f(x,y) = (x+e^y)^7\) |
| 14. \(f(x,y) = x^4 + 4x^3y - 6x^2y^2 - 4xy^3 + y^4\) |
| 15. \(f(x,y) = \sqrt{x+\sqrt{y}}\) |
| 16. \(f(x,y) = x^2y^3-4x^3\) |
Here is a table showing the function \(A(t,r)\)
| \(\overset{t}{\downarrow} \ r\rightarrow\) |
.03 |
.04 |
.05 |
.06 |
.07 |
|
1 |
30.45 |
40.81 |
51.27 |
61.84 |
72.51 |
|
2 |
61.84 |
83.29 |
105.17 |
127.50 |
150.27 |
|
3 |
94.17 |
127.50 |
161.83 |
197.22 |
233.68 |
a. Estimate \(A_t (2, .05)\).
b. Estimate \(A_r (2, .05)\)
c. Use your answers to parts a and b to estimate the value of \(A (2.5, .054)\)
d. The values in the table came from \(A(t,r)=1000 (e^{rt} - 1)\), which shows the interest earned if 1000 dollars is deposited in an account earning r annual interest, compounded continuously, and left there for \(t\) years. How close are your estimates from parts a, b, and c?
18. Here is a table showing values for the function \(H(t,h)\).
| \(\overset{t}{\downarrow} \ h\rightarrow\) |
100 |
150 |
200 |
|
0 |
100 |
150 |
200 |
|
1 |
110.1 |
160.1 |
210.1 |
|
2 |
110.4 |
160.4 |
210.4 |
|
3 |
100.9 |
150.9 |
200.9 |
|
4 |
81.6 |
131.6 |
181.6 |
|
5 |
52.5 |
102.5 |
152.5 |
a. Estimate the value of \(\frac{\partial H}{dt}\) at (3, 150).
b. Estimate the value of \(\frac{\partial H}{dh}\) at (3, 150).
c. Use your answers to parts a and b to estimate the value of \(H(2.6, 156)\).
d. The values in the table came from \(H (t,h) = h + 15t - 4.9t^2\), which gives the height in meters above the ground after \(t\) seconds of an object that is thrown upward from an initial height of \(h\) meters with an initial velocity of 15 meters per second. How close are your estimates from parts a, b, and c?
Given the function \(f(x,y) = x^2\sqrt{y}\)
a. Calculate \(f(2,4)\), \(f_x (2,4)\), and \(f_y(2,4)\)
b. Use your answers from part \(a\) to estimate \(f(1.9, 4.1)\)
Given the function \(f(x,y) = \ln (10 - x^2 - y)\)
a. Calculate \(f(2,5)\), \(f_x (2,5)\), and \(f_y(2,5)\)
b. Use your answers from part \(a\) to estimate \(f(1.8,4.8)\)
In problems 21 - 26, use the contour plot shown to estimate the desired value.
| 21. \(f_x(1,-5)\) |
| 22. \(f_x(-5,2)\) |
| 23. \(f_x(5,5)\) |
| 24. \(f_x(0,0)\) |
| 25. \(f_y(1,-5)\) |
| 26. \(f_y(-5,2)\) |