Q: A population of bacteria grows according to $$P(t) = 2t^5$ - t^3$ + 4. $$What is the rate of growth at $$t = 2$$?

$$P'(2) = 2 \times 5 \times 2^4$ - 3 \times 2^2$ = 160 - 12 = 148$$

Question 1

Given the function $$f(x) = 20x^5$ - 4x^4$ + 8x^3$ - 4x$$, find $$f'(x). $$Which of the following is the correct derivative?

Accepted Answer

$$f'(x) = 100x^4$ - 16x^3$ + 24x^2$ - 4$$

Question 2

Suppose $$f(x) = 3x^9$. What is f$$'(x)$$?

Accepted Answer

$$f'(x) = 27x^8$

Question 3

A pharmaceutical company models the rate of change of a drug's concentration in the bloodstream as $$C(t) = 4t^7$. What is the instantaneous rate of change at t = 2$ hours?

Accepted Answer

$$448$$

Question 4

Which of the following best describes the meaning of the derivative $$f'(x) at a $$point $$x = a$$?

Accepted Answer

The instantaneous rate of change of $$f(x)$$ with respect to $$x at$$ $$x = a$$

Question 5

Given $$f(x) = 10x^4$ - 4x^3$ + 8x^2$ - 4x$$, what is the value of $$f''(x)$$?

Accepted Answer

$$f''(x) = 120x^2$ - 24x + 16$$

Question 6

If $$f(x) = ax^n$$, where $$a$$ and $$n$$ are constants, what is the general formula for $$f'(x)$$?

Accepted Answer

$$f'(x) = a n x$$^{n-1}$$

Question 7

A car's position is modeled by $$s(t) = 5t^4$ - 2t^2$ + 3. $$What is the car's velocity at time $$t = 1$$?

Accepted Answer

$$v(1) = 18$$

Question 8

Which of the following functions has a derivative of zero for all $$x$$?

Accepted Answer

$$f(x) = 7$$

Question 9

If $$f(x) = 4x^7$, what is the equation of the tangent line to f(x$$)$$ at x = 1$?

Accepted Answer

$$y = 28x - 24$$

Question 10

A population of bacteria grows according to $$P(t) = 2t^5$ - t^3$ + 4. $$What is the rate of growth at $$t = 2$$?

Accepted Answer

$$P'(2) = 2 	imes 5 	imes 2^4$ - 3 	imes 2^2$ = 160 - 12 = 148$$

Question 11

Which of the following statements is TRUE about the power rule for differentiation?

Accepted Answer

The power rule applies to all real exponents $$n$$

Question 12

If $$f(x) = 7x^3$ - 2x^2$ + x - 5$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = 21x^2$ - 4x + 1$$

Question 13

A chemical reaction's rate is modeled by $$R(t) = 6t^2$ - 4t + 2. $$What is the rate of change at $$t = 3$$?

Accepted Answer

$$R'(3) = 32$$

Question 14

If $$f(x) = x^n$$, which of the following is the correct derivative?

Accepted Answer

$$f'(x) = n x$$^{n-1}$$

Question 15

Suppose $$f(x) = 5x^4$ - 3x^2$ + 2. $$What is the slope of the tangent line at $$x = -1$$?

Accepted Answer

$$f'(-1) = -17$$

Question 16

Which of the following is the correct derivative of $$f(x) = 2x^6$ - 5x^3$ + 4x$$?

Accepted Answer

$$f'(x) = 12x^5$ - 15x^2$ + 4$$

Question 17

A company's profit is modeled by $$P(x) = 8x^3$ - 6x^2$ + 2x - 1. $$What is the marginal profit function?

Accepted Answer

$$P'(x) = 24x^2$ - 12x + 2$$

Question 18

If $$f(x) = 9x^2$ - 4x + 7$$, what is $$f'(2)$$?

Accepted Answer

$$f'(2) = 28$$

Question 19

Which of the following is NOT a correct application of the power rule?

Accepted Answer

For $$f(x) = \ln(x)$$, $$f'(x) = x$$^{-1}$$

Question 20

A ball is thrown upward and its height is given by $$h(t) = -16t^2$ + 32t + 5. $$What is the velocity at $$t = 1$$ second?

Accepted Answer

$$v(1) = 0$$

Step-by-Step Guidance for Derivatives Quiz 1 (Calculus)

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