Q: A function $$f(x) is $$increasing on an interval $$(a, b) if$$:

$$f'(x) \u003e 0$$ \text{ for all } x \in (a, b)$

Q: Given $$L(t) = -37.50 + 3.71t - 6.33 \times 10$$^{-4}$$ t^3$, representing the femur length (in mm) of a fetus as a function of age t$ (in weeks), what is the rate of growth $$\frac{dL}{dt}$$ at t = 20$ weeks?

$$3.71 - 3 \times 6.33 \times 10$$^{-4}$$ \times 20^2$

Q: A function $$f(x) is $$decreasing on an interval $$(b, c) if$$:

$$f'(x) \u003c 0$$ \text{ for all } x \in (b, c)$

Question 1

Let $$s(t) = -16t^2$ + 100t + 85$$ represent the height (in feet) of a potato launched vertically upward from an 85-foot building with an initial velocity of 100 ft/s. What is the velocity of the potato when it hits the ground?

Accepted Answer

$$-32(7.0081) + 100$$

Question 2

Given the position function $$s(t) = 4t^6$ - 5t^2$ + t - 6$$, what is the acceleration at $$t = 2$$ seconds?

Accepted Answer

$$120 \cdot 2^4$ - 10$$

Question 3

A function $$f(x) is $$increasing on an interval $$(a, b) if$$:

Accepted Answer

$$f'(x) \u003e 0$$ 	ext{ for all } x \in (a, b)$

Question 4

If the derivative of a function is negative on an interval $$(a, b)$$, what can you conclude about the function on that interval?

Accepted Answer

The function is decreasing on $$(a, b)$$

Question 5

Given $$L(t) = -37.50 + 3.71t - 6.33 	imes 10$$^{-4}$$ t^3$, representing the femur length (in mm) of a fetus as a function of age t$ (in weeks), what is the rate of growth $$\frac{dL}{dt}$$ at t = 20$ weeks?

Accepted Answer

$$3.71 - 3 	imes 6.33 	imes 10$$^{-4}$$ 	imes 20^2$

Question 6

Suppose the velocity of an object is given by $$v(t) = s'(t). $$What does $$|v(t)|$$ represent?

Accepted Answer

The speed of the object at time $$t$$

Question 7

A function $$f(x) is $$decreasing on an interval $$(b, c) if$$:

Accepted Answer

$$f'(x) \u003c 0$$ 	ext{ for all } x \in (b, c)$

Question 8

If $$s(t) is $$the position function of an object, which of the following gives the acceleration at time $$t$$?

Accepted Answer

$$a(t) = s''(t)$$

Question 9

A ball is thrown upward from a height of 50 ft with an initial velocity of 80 ft/s. Its height after $$t$$ seconds is $$s(t) = -16t^2$ + 80t + 50. $$How long does it take to hit the ground?

Accepted Answer

$$t = \frac{-80 \pm \sqrt{80^2$ - 4(-16)(50)}}{2(-16)}$$

Question 10

Given $$s(t) = -16t^2$ + 100t + 85$$, what is the initial velocity of the object?

Accepted Answer

$$100 ft/s$$

Question 11

If $$f(x) is $$increasing on $$(a, b)$$ and decreasing on $$(b, c)$$, what can you say about $$f at$$ $$x = b$$?

Accepted Answer

$$f$$ has a local maximum at $$x = b$$

Question 12

For the function $$f(x)$$ shown in the graph, the slope of the tangent line is positive where:

Accepted Answer

$$f is $$increasing

Question 13

If the acceleration of an object is negative, what does this indicate about the motion of the object?

Accepted Answer

The velocity is decreasing

Question 14

Given $$s(t) = 4t^6$ - 5t^2$ + t - 6$$, what is the velocity at $$t = 2$$?

Accepted Answer

$$24 \cdot 2^5$ - 10 \cdot 2 + 1$$

Question 15

A fetus's femur length is modeled by $$L(t) = -37.50 + 3.71t - 6.33 	imes 10$$^{-4}$$ t^3$. What can you say about the rate of growth as t$ increases?

Accepted Answer

The rate of growth decreases as $$t$$ increases

Question 16

If $$f'(x) = 0$$ for all $$x in an $$interval, what can you conclude about $$f(x) on $$that interval?

Accepted Answer

$$f(x) is $$constant on that interval

Question 17

Which of the following best describes the relationship between the sign of the derivative and the behavior of a function?

Accepted Answer

If $$f'(x) \u003e 0$$, $$f(x) is $$increasing; if $$f'(x) \u003c 0$$, $$f(x) is $$decreasing

Position, Velocity, Acceleration, and Increasing/Decreasing Functions in Calculus

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