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: Suppose the femur length $$L (in mm) of a $$fetus is modeled by $$L = -37.50 + 3.71t - 6.33 \times 10$$^{-4}$$ t^3$, where t$ is the age in weeks. What is the rate of growth $$\frac{dL}{dt}$$ at t = 20$ weeks?

$$\frac{dL}{dt} = 3.71 - 3 \times 6.33 \times 10$$^{-4}$$ \times 20^2$ = 3.71 - 0.7596 = 2.9504$$

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

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

Question 1

Let $$s(t) be $$the position function of an object moving vertically, given by $$s(t) = -16t^2$ + 100t + 85. $$What is the velocity of the object at $$t = 2$$ seconds?

Accepted Answer

$$v(2) = -32(2) + 100 = 36$$

Question 2

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

Accepted Answer

$$a(2) = 120(2^4$) - 10 = 1910$$

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

Suppose the femur length $$L (in mm) of a $$fetus is modeled by $$L = -37.50 + 3.71t - 6.33 	imes 10$$^{-4}$$ t^3$, where t$ is the age in weeks. What is the rate of growth $$\frac{dL}{dt}$$ at t = 20$ weeks?

Accepted Answer

$$\frac{dL}{dt} = 3.71 - 3 	imes 6.33 	imes 10$$^{-4}$$ 	imes 20^2$ = 3.71 - 0.7596 = 2.9504$$

Question 5

If the velocity of an object is given by $$v(t) = s'(t)$$, and $$v(t) is $$negative on an interval, what can you conclude about the motion of the object?

Accepted Answer

The object is moving in the negative direction on that interval.

Question 6

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

Accepted Answer

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

Question 7

A potato is launched vertically upward from an 85-foot building with an initial velocity of 100 ft/s. Its position is given by $$s(t) = -16t^2$ + 100t + 85. $$How long does it take for the potato to hit the ground?

Accepted Answer

$$t = \frac{-100 + \sqrt{100^2$ - 4(-16)(85)}}{2(-16)} \approx 7.0081$$ seconds

Question 8

Given $$f(x) is $$increasing on $$(a, b)$$ and decreasing on $$(b, c)$$, what can you say about $$f'(b) if$$ $$f is $$differentiable at $$b$$?

Accepted Answer

$$f'(b) = 0$$

Question 9

If the acceleration $$a(t) of an $$object is negative while its velocity $$v(t) is $$positive, what is happening to the object's speed?

Accepted Answer

The object is slowing down.

Question 10

Suppose $$f(x) is a $$function such that $$f'(x) \u003e 0$$ for $$x in$$ $$(1, 4)$$, $$f'(x)  0$$ for $$x in$$ $$(7, 10). $$Which of the following best describes the behavior of $$f(x)$$?

Accepted Answer

$$f(x)$$ increases on $$(1, 4)$$, decreases on $$(4, 7)$$, and increases on $$(7, 10)$$

Question 11

A function $$f(x)$$ has a positive slope for all $$x in$$ $$(a, b). $$What does this imply about the tangent lines to the graph of $$f(x) on$$ $$(a, b)$$?

Accepted Answer

The tangent lines all have positive slopes.

Question 12

Given the position function $$s(t) = -16t^2$ + 100t + 85$$, what is the acceleration of the object at any time $$t$$?

Accepted Answer

$$a(t) = -32$$

Question 13

If $$f(x) is $$decreasing on $$(a, b)$$, which of the following is true for any $$x_1$$, $$x_2 in$$ $$(a, b)$$ with $$x_1$$ \u003c $$x_2$$?

Accepted Answer

$$f(x_1$) \u003e f(x_2$)$$

Question 14

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

Accepted Answer

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

Question 15

If the slope of the tangent line to $$f(x) at$$ $$x = a is $$negative, what does this indicate about $$f(x) at$$ $$x = a$$?

Accepted Answer

$$f(x) is $$decreasing at $$x = a$$

Question 16

Suppose $$s(t) is $$the position of a car in meters at time $$t in $$seconds. If $$s'(t) = 0 at$$ $$t = 5$$, what does this mean physically?

Accepted Answer

The car is momentarily at rest at $$t = 5$$ seconds.

Question 17

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 18

A ball is thrown upward and its position is given by $$s(t) = -16t^2$ + 64t + 10. $$What is the maximum height reached by the ball?

Accepted Answer

$$s(2) = -16(2^2$) + 64(2) + 10 = 74$$ feet

Calculus Study Notes: Position, Velocity, Acceleration, and Increasing/Decreasing Functions

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