Applications Of Differential Calculus — Quiz 3

Questions in this quiz (10)

1. The sum of the squares of two positive numbers is $$180$$. Their minimum product is:

   • $$90$$
   • $$45$$
   • $$0$$
   • There is no minimum

2. If $$h$$ is a small number, the local linear approximation for$$\sqrt[3]{64+h}$$ is:

   • $$4+\frac{h}{48}$$
   • $$4+\frac{h}{12}$$
   • $$4+\frac{h}{24}$$
   • $$4-\frac{h}{48}$$

3. If $$f(x)=xe^{-2x}$$ then at $$x=0$$:

   • $$f$$ is increasing
   • $$f$$ is decreasing
   • $$f$$ has a relative maximum
   • $$f$$ has a relative minimum

4. A function $$f$$ has a derivative for each $$x$$ such that $$|x|$$ < $$3$$ and has a local maximum at $$(1,4)$$. Which statement below must be true?

   • $$f'(1)=0$$
   • $$f''(1)$$ exists
   • $$f'(x)$$ > $$0$$ if $$x$$ < $$1$$ and $$f'(x)$$ < $$0$$ if $$x$$ > $$1$$
   • None of the preceding is necessarily true

5. The height of a rectangular box is $$12$$ $$in$$. Its length increases at the rate of $$3$$ $$in/sec$$, and its width decreases at the rate of $$2$$ $$in/sec$$. When the length is $$10$$ $$in$$ and the width is $$8$$ $$in$$, the rate, in cubic inches per second, at which the volume of the box is changing is:

   • $$168$$
   • $$48$$
   • $$-96$$
   • $$-168$$

6. The tangent to the curve$$x^3+xy^2+2y=4$$ at the point $$(1,1)$$ has slope:

   • $$1$$
   • $$-\frac{3}{2}$$
   • $$-\frac{1}{2}$$
   • $$-1$$

7. If $$f(x)=ax^4+bx^2$$ and $$ab$$ < $$0$$ then:

   • the curve has exactly one horizontal tangent
   • the curve is concave up for all $$x$$
   • the curve has an inflection point
   • none of the preceding is necessarily true

8. A function $$f$$ is continuous and differentiable on the interval $$[0,4]$$, where $$f'$$ is negative but $$f''$$ is positive. Which table could represent points on $$f$$?

9. Given the function $$f(x)=\sqrt[3]{x}$$. It is known that $$f(27)=3$$. Using the tangent line to the graph of $$f(x)$$, approximately how much less than $$3$$ is $$\sqrt[3]{26}$$?

   • $$\frac{1}{81}$$
   • $$\frac{1}{27}$$
   • $$\frac{1}{9}$$
   • $$\frac{2}{9}$$

10. The best linear approximation for $$f(x)=\sin x$$ near $$x=\frac{\pi}{6}$$ is:

    • $$\frac{1}{2}+\frac{\sqrt{3}}{2}\left(x-\frac{\pi}{6}\right)$$
    • $$\frac{1}{2}+\left(x-\frac{\pi}{6}\right)$$
    • $$\frac{1}{2}+\frac{1}{2}\left(x-\frac{\pi}{6}\right)$$
    • $$\frac{1}{2}+\sqrt{3}\left(x-\frac{\pi}{6}\right)$$