Question 1

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

Accepted Answer

$$f'(x) = 6x^2$ + 6x - 2$$

Question 2

A quadratic polynomial $$f(x) = ax^2$ + bx + c$$ satisfies $$f(0) = 5$$, $$f'(0) = 3$$, and $$f''(x) = 6$$ for all $$x. $$What is $$f(x)$$?

Accepted Answer

Question 3

If the concentration of an antibiotic in the bloodstream $$t$$ hours after injection is given by $$C(t) = \frac{20t}{t^2$ + 1}$$, what is the maximum concentration?

Accepted Answer

$$10$$

Question 4

For the function $$f(x) = 2x^3$ + 3x^2$ - 2x$$, at what $$x-$$value does the function have a point of inflection?

Accepted Answer

$$x = -1$$

Question 5

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

Accepted Answer

$$y = 7x - 2$$

Question 6

Suppose $$f(x) = 2x^3$ + 3x^2$ - 2x. $$What is the value of $$f''(2)$$?

Accepted Answer

$$18$$

Question 7

A patient receives an antibiotic injection. The concentration in the bloodstream is $$C(t) = \frac{20t}{t^2$ + 1}. At $$what time $$t is $$the concentration increasing most rapidly?

Accepted Answer

$$t = 1$$

Question 8

Given $$f(x) = ax^2$ + bx + c$$ with $$f(0) = 5$$, $$f'(0) = 3$$, and $$f''(x) = 6$$, what are the values of $$a$$, $$b$$, and $$c$$?

Accepted Answer

$$a = 3, b = 3, c = 5$$

Question 9

For the function $$C(t) = \frac{20t}{t^2$ + 1}$$, what is $$C'(t)$$?

Accepted Answer

$$C'(t) = \frac{20(1 - t^2$)}{(t^2$ + 1)^2$}$$

Question 10

A quadratic $$f(x) = ax^2$ + bx + c$$ has $$f(0) = 5$$, $$f(1) = 10$$, and $$f(2) = 25. $$What is $$a$$?

Accepted Answer

$$a = 5$$

Question 11

Given $$f(x) = 2x^3$ + 3x^2$ - 2x$$, what is the $$y-$$intercept of the graph?

Accepted Answer

$$-2$$

Question 12

If $$f(x) = 2x^3$ + 3x^2$ - 2x$$, what is the value of $$x$$ where $$f'(x) = 0$$?

Accepted Answer

$$x = \frac{-3 \pm \sqrt{21}}{6}$$

Question 13

A medication is modeled by $$C(t) = \frac{20t}{t^2$ + 1}. $$What is the long-term behavior of $$C(t) as$$ $$t 	o \infty$$?

Accepted Answer

$$C(t) 	o 0$$

Question 14

Given $$f(x) = 2x^3$ + 3x^2$ - 2x$$, what is the concavity at $$x = 0$$?

Accepted Answer

Concave up

Question 15

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

Accepted Answer

$$-5$$

Question 16

For $$C(t) = \frac{20t}{t^2$ + 1}$$, what is the time $$t$$ when the concentration is half its maximum value?

Accepted Answer

$$t = 2.618$$

Calculus Study Notes: Quadratic Polynomials, Derivatives, and Applications

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