Question 1

Given the function $$f(x) = x^4$ - 3x^3$ + 2x - 5$$, what is the third derivative $$f'''(x)$$?

Accepted Answer

$$24x - 18$$

Question 2

A company’s profit function is $$P(x) = -2x^3$ + 15x^2$ + 36x - 50 (in $$thousands of dollars), where $$x is $$the number of units produced (in hundreds). What does $$P''(x)$$ represent in this context?

Accepted Answer

The rate of change of marginal profit with respect to production

Question 3

Find the fourth derivative of $$g(x) = x^5$ - 3x^4$ + 2x^2$.

Accepted Answer

$$120x - 72$$

Question 4

Given $$h(x) = 2x^5$ - x^4$ + 3x^2$ - 7$$, what is $$h'''(x)$$?

Accepted Answer

$$240x^2 - 24x$

Question 5

A company’s cost function is $$C(x) = 0.01x^3$ - 0.3x^2$ + 5x + 100. $$What is the marginal cost function $$C'(x)$$?

Accepted Answer

$$0.03x^2 - 0.6x + 5$

Question 6

For the revenue function $$R(x) = 50x - 0.02x^3$, what is R$$''(x)$$?

Accepted Answer

$$-0.12x$$

Question 7

Suppose $$y = (x^2$ + 1)(x^2$ - 1). $$What is $$\frac{d^2$y}{dx^2$}$$?

Accepted Answer

$$12x^2 - 4$

Question 8

A population model is $$P(t) = 50000 + 2000t^2$ - 10t^3$, where t$ is years after 2020. At what value of t$ does the population growth rate start to decrease?

Accepted Answer

Question 9

Given $$f(x) = x^6$ - 2x^3$ + x$$, what is $$f^{(4)}(x$$)$$?

Accepted Answer

$$360x^2 - 24$

Question 10

If $$f(x) = ax^3$ + bx^2$ + cx + d$$ and $$f''(2) = 12$$, $$f'''(1) = 18$$, what are the values of $$a$$ and $$b$$?

Accepted Answer

$$a = 3, b = 0$$

Question 11

For the function $$g(x) = \frac{x^4$ - 2x^2$}{x}$$, what is $$g''(x)$$?

Accepted Answer

$$12x - 4x$$^{-3}$$

Question 12

A company’s cost function is $$C(x) = 0.01x^3$ - 0.3x^2$ + 5x + 100. At $$what production level $$x is$$ $$C''(x) = 0$$?

Accepted Answer

$$x = 10$$

Question 13

The revenue function for a product is $$R(x) = 50x - 0.02x^3$. For which values of x$ is the revenue function concave down?

Accepted Answer

$$x \u003e 0$$

Question 14

A manufacturing company’s efficiency function is $$E(t) = -t^4$ + 8t^3$ - 18t^2$ + 12t + 50. At $$what value of $$t is $$the rate of efficiency change maximized?

Accepted Answer

$$t = 3$$

Question 15

For the function $$f(x) = x^4$ - 8x^3$ + 12x^2$, what is the first value of n$ such that f$$^{(n)}$$(x) = 0$$ for all $$x$$?

Accepted Answer

$$n = 4$$

Question 16

Given the population model $$P(t) = 50000 + 2000t^2$ - 10t^3$, what is the acceleration of population change P$$''(t)$$?

Accepted Answer

$$4000 - 60t$$

Question 17

For the efficiency function $$E(t) = -t^4$ + 8t^3$ - 18t^2$ + 12t + 50$$, what is the concavity at $$t = 2$$ hours?

Accepted Answer

Concave up

Question 18

If $$f(x) = 2x^3$ - 1/x$$, what is $$f''(x)$$?

Accepted Answer

$$12x - 2x$$^{-3}$$

Question 19

Which of the following best describes the economic meaning of a positive second derivative of a cost function $$C''(x) \u003e 0$$?

Accepted Answer

The marginal cost is increasing as production increases

Question 20

Suppose a business finds that $$C''(x) = 0 at$$ $$x = 10. $$What does this indicate about the cost function at $$x = 10$$?

Accepted Answer

The marginal cost is neither increasing nor decreasing at $$x = 10$$

Step-by-Step Guidance for Higher Derivatives and Applications (Calculus)

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