Q: A sphere is growing at a rate of $$50\ \text{cm}^3/\text{s}. At $$what rate is the radius increasing when the radius is $$5 cm$$?

$$\frac{1}{2\pi}$$\text{ cm/s}$

Q: As an ice cube melts, each side changes at $$-0.3 mm/$$min. When each side is $$0.9 cm$$, what is the rate of change of the ice cube's volume?

$$-0.0243$$\text{ cm}^3/\text{min}$

Question 1

Given the position function $$s(t) = 9t - t^2$ for 0$$ \leq t \leq 3$$, what is the average velocity of the object over the interval $$[0,3]$$?

Accepted Answer

$$3$$	ext{ m/s}$

Question 2

A projectile is shot vertically upward from a point $$5 m $$above ground with initial velocity $$20 m/s. $$Its height is given by $$s(t) = 5 + 20t - 4.9t^2$. What is the vertical velocity at t = 3$ seconds?

Accepted Answer

$$-14.7$$	ext{ m/s}$

Question 3

Given the velocity function $$v(t) = t^2$ - 3t^2$ + 2t$$, what is the instantaneous acceleration at $$t = 3$$?

Accepted Answer

$$-16$$	ext{ m/s}^2$

Question 4

For the position function $$s(t) = 12t - t^2$, what is the acceleration of the object at t = 5$ seconds?

Accepted Answer

$$-2$$	ext{ m/s}^2$

Question 5

Given the equation $$5x^2$$ + $$y^3 = 12$, find $$\frac{dy}{dx}$$ using implicit differentiation.

Accepted Answer

$$\frac{-10x}{3y^2$}$$

Question 6

If $$xy = \sin(y)$$, what is $$\frac{dy}{dx}$$ using implicit differentiation?

Accepted Answer

$$\frac{\sin(y) - y}{x - \cos(y)}$$

Question 7

A sphere is growing at a rate of $$50\ 	ext{cm}^3/	ext{s}. At $$what rate is the radius increasing when the radius is $$5 cm$$?

Accepted Answer

$$\frac{1}{2\pi}$$	ext{ cm/s}$

Question 8

A right triangle has a base of $$10 cm $$and a height of $$12 cm. $$The height is decreasing at $$0.4 cm/$$min. At what rate is the area decreasing?

Accepted Answer

$$2$$	ext{ cm}^2/	ext{min}$

Question 9

If the perimeter of a rectangle is fixed at $$30 cm $$and the length $$L is $$increasing at $$2 cm/s$$, for what value of $$L$$ does the area start to decrease?

Accepted Answer

$$7.5$$	ext{ cm}$

Question 10

As an ice cube melts, each side changes at $$-0.3 mm/$$min. When each side is $$0.9 cm$$, what is the rate of change of the ice cube's volume?

Accepted Answer

$$-0.0243$$	ext{ cm}^3/	ext{min}$

Question 11

A 15-foot plank leans against a vertical pole. The top slides down at $$2 in/s. $$How fast is the bottom moving away when it is $$8 ft $$from the base?

Accepted Answer

$$1.07$$	ext{ in/s}$

Question 12

Two cars leave the same intersection: Car A east at $$60 mi/h$$, Car B north at $$40 mi/h. $$Car A leaves at 2pm, Car B at 2:30pm. At 3pm, how fast is the distance between them changing?

Accepted Answer

$$32$$	ext{ mi/h}$

Question 13

If $$f(x) = x^2$ and the linearization at x = 1$ is L(x) = 2x - 1$, what is the approximate value of f(1.05$$)$$?

Accepted Answer

$$1.10$$

Question 14

Given $$f(x) = x^2$ + 1$$, use the linearization at $$a = 5 to $$approximate $$f(5.1).$$

Accepted Answer

$$26.1$$

Question 15

Given $$f(x) = \sqrt{x} + 12$$, use the linearization at $$a = 16 to $$approximate $$f(16.01).$$

Accepted Answer

$$16.00125$$

Question 16

Given $$y = \sqrt{2x + 1}$$, $$\frac{dx}{dt} = 12$$, and $$x = \frac{15}{7}$$, what is $$\frac{dy}{dt}$$?

Accepted Answer

$$8$$

Question 17

Given $$x^2$$ + $$y^2$$ - $$3z^2 = 125$, $$\frac{dx}{dt} = 3$$, x = 2$, y = 4$, z = 1$, what is $$\frac{dz}{dt}$$?

Accepted Answer

$$3$$

Calculus Study Guide: Derivatives in Motion, Implicit Differentiation, Related Rates, and Linear Approximation

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