Q: A cube grows at a rate of $$2 cm$^3/s. $$How fast is the side of the cube growing when it has a length of $$0.8 cm$$?

$$\frac{2}{3 \times (0.8)^2$} cm/s$$

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

$$0 m/s$$

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

$$-9.4 m/s$$

Question 3

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

Accepted Answer

$$-16 m/s$^2$$

Question 4

The position of an object is given by $$s(t) = 12t - t^2$. What is the acceleration of the object at t = 5$ seconds?

Accepted Answer

$$-2 m/s$^2$$

Question 5

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

Accepted Answer

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

Question 6

A cube grows at a rate of $$2 cm$^3/s. $$How fast is the side of the cube growing when it has a length of $$0.8 cm$$?

Accepted Answer

$$\frac{2}{3 	imes (0.8)^2$} cm/s$$

Question 7

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

Accepted Answer

$$\frac{50}{4\pi (5)^2$} cm/s$$

Question 8

If $$y = \sqrt{2x + 1}$$, and $$\frac{dx}{dt} = 12$$ when $$x = \frac{15}{7}$$, what is $$\frac{dy}{dt} at $$that instant?

Accepted Answer

$$\frac{12}{\sqrt{2 \cdot \frac{15}{7} + 1}}$$

Question 9

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

Accepted Answer

$$-2 cm$^2/$$min

Question 10

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

Accepted Answer

$$11.2$$

Question 11

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

Accepted Answer

$$\frac{7}{12}$$

Question 12

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

Accepted Answer

$$\frac{16}{15} ft/s$$

Question 13

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

Accepted Answer

$$4.00125$$

Question 14

Given $$xy = \sin(y)$$, find $$\frac{dy}{dx}$$ using implicit differentiation.

Accepted Answer

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

Question 15

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

Accepted Answer

$$7.5 cm$$

Question 16

If $$y = \sqrt{49 - x^2$}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{-x}{\sqrt{49 - x^2$}}$$

Question 17

Given the position function $$s(t) = \frac{30}{t + 5}$$ for $$-4 \leq t \leq 0$$, what is the average velocity over the interval?

Accepted Answer

$$-1.5 m/s$$

Question 18

A car travels east at $$60 mi/h $$and another travels north at $$40 mi/h. If $$the eastbound car leaves at 2pm and the northbound at 2:30pm, at what rate is the distance between them changing at 3pm?

Accepted Answer

$$\frac{100}{\sqrt{60^2$ + 40^2$}} mi/h$$

Question 19

If an ice cube melts so that each side changes at $$-3 mm/$$min, what is the rate of change of the volume when each side is $$0.9 cm$$?

Accepted Answer

$$-0.243 cm$^3/$$min

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

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