Q: Suppose $$y = x^2$ + 4x + 4. $$Use differentials to approximate the change in $$y$$ when $$x$$ increases from $$2 to$$ $$2.05.$$

$$dy = 8.2 \times 0.05 = 0.41$$

Question 1

Let $$f(x) = x^3$ - 3x^2$ + 2. $$Using the First Derivative Test, at which $$x-$$value does $$f(x)$$ have a relative maximum?

Accepted Answer

$$x = -1$$

Question 2

Suppose $$f(x) is $$continuous and differentiable on $$(-\infty, \infty)$$ and $$f'(x) \u003e 0$$ for all $$x. $$Which of the following statements is true?

Accepted Answer

$$f(x) is $$increasing everywhere.

Question 3

A company’s cost function is $$C(x) = x^2$ + 40x + 200. $$What is the marginal cost when $$x = 10$$?

Accepted Answer

$$C'(10) = 2x + 40 = 60$$

Question 4

Given $$f(x) = \dfrac{8}{x^2$ - 4}$$, which of the following is a vertical asymptote of $$f(x)$$?

Accepted Answer

$$x = 2$$

Question 5

For the function $$f(x) = x^2$ e$$^{-x}$$, what are the coordinates of the relative maximum?

Accepted Answer

$$(2, 4e$$^{-2}$$)$$

Question 6

A box with a square base and no top is to be made from a $$10 in. by$$ $$10 in. $$piece of cardboard by cutting equal squares from each corner and folding up the sides. What value of $$x ($$the side of the square cut out) maximizes the volume?

Accepted Answer

$$x = 1.667$$

Question 7

The elasticity of demand for a product is given by $$E(p) = \dfrac{5p}{176 - 5p}. At $$what price is the total revenue maximized?

Accepted Answer

$$p = 17.6$$

Question 8

Suppose $$f(x) is $$continuous on $$[1, 5]$$ and $$f(x) = 4x - x^2$. What is the absolute minimum value of f(x$$)$$ on $$[1, 5]$$?

Accepted Answer

$$-5 at$$ $$x = 5$$

Question 9

If $$y = x^3$ and $$\dfrac{dx}{dt} = -4$$ when x = 9$, what is $$\dfrac{dy}{dt}$$ at that moment?

Accepted Answer

$$-108$$

Question 10

Given the demand function $$D(p) = 120 - 20p$$, what is the elasticity of demand at $$p = 2$$?

Accepted Answer

$$E(2) = 0.5$$

Question 11

For the function $$f(x) = x^3$ - 3x^2$ + 2$$, what is the second derivative $$f''(x)$$?

Accepted Answer

$$6x - 6$$

Question 12

A company’s revenue function is $$R(x) = 1000x - 2x^2$. If the company is increasing production at 10$ units per week when x = 400$, what is the rate of change of revenue at that moment?

Accepted Answer

$$2000$$ per week

Question 13

Which of the following best describes the concavity of $$f(x) if$$ $$f''(x) \u003e 0$$ for all $$x in an $$interval?

Accepted Answer

$$f(x) is $$concave up on the interval.

Question 14

Suppose $$y = x^2$ + 4x + 4. $$Use differentials to approximate the change in $$y$$ when $$x$$ increases from $$2 to$$ $$2.05.$$

Accepted Answer

$$dy = 8.2 	imes 0.05 = 0.41$$

Question 15

If $$f(x) = x^3$ - 3x^2$ + 2$$, what is the equation of the tangent line at $$x = 1$$?

Accepted Answer

$$y = -1(x - 1) + 0$$

Question 16

A restaurant supplier services a circular area whose radius $$r is $$increasing at $$2 mi/$$year when $$r = 5 mi. $$How fast is the area increasing at that moment?

Accepted Answer

$$20\pi sq mi/$$year

Question 17

Given $$y = x^2$ + 4x + 4$$, what is the linearization at $$x = 2$$?

Accepted Answer

$$y = 8(x - 2) + 16$$

Question 18

If $$y = x^3$ and $$\dfrac{dx}{dt} = -4$$ when x = 9$, what is $$\dfrac{dy}{dt}$$ at that moment?

Accepted Answer

$$-108$$

Question 19

Suppose $$f(x) = x^2$ e$$^{-x}$$. What is the x$-coordinate of the relative minimum?

Accepted Answer

$$x = 0$$

Calculus and Its Applications – Chapter 3: Applications of Differentiation

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