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### Section 4.12 : Differentials

For problems 1 – 5 compute the differential of the given function.

1. $$f\left( x \right) = 3{x^6} - 8{x^3} + {x^2} - 9x - 4$$
2. $$u = {t^2}\cos \left( {2t} \right)$$
3. $$y = {{\bf{e}}^{\cos \left( z \right)}}$$
4. $$g\left( z \right) = \sin \left( {3z} \right) - \cos \left( {1 - z} \right)$$
5. $$R\left( x \right) = \sqrt{{6x + {{\bf{e}}^{ - x}}}}$$
6. Compute $$dy$$ and $$\Delta y$$ for $$y = \sin \left( x \right)$$ as x changes from 6 radians to 6.05 radians.
7. Compute $$dy$$and $$\Delta y$$ for $$y = \ln \left( {{x^2} + 1} \right)$$ as x changes from -2 to -2.1.
8. Compute $$dy$$ and $$\Delta y$$ for $$\displaystyle y = \frac{1}{{x - 2}}$$ as x changes from 3 to 3.02.
9. Compute $$dy$$ and $$\Delta y$$ for $$y = x\,{{\bf{e}}^{\frac{1}{4}x}}$$ as x changes from -10 to -9.99.
10. The sides of a cube are found to be 6 feet in length with a possible error of no more than 1.5 inches. What is the maximum possible error in the surface area of the cube if we use this value of the length of the side to compute the surface area?
11. The radius of a circle is found to be 7 cm in length with a possible error of no more than 0.04 cm. What is the maximum possible error in the area of the circle if we use this value of the radius to compute the area?
12. The radius of a sphere is found to be 22 cm in length with a possible error of no more than 0.07 cm. What is the maximum possible error in the volume of the sphere if we use this value of the radius to compute the volume?
13. The radius of a sphere is found to be ½ foot in length with a possible error of no more than 0.03 inches. What is the maximum possible error in the surface area of the sphere if we use this value of the radius to compute the surface area?