Question 1

A spherical balloon is being inflated so that its volume increases at a rate of $$100 	ext{ cm}^3/	ext{min}. $$How fast is the radius increasing when the radius is $$5 	ext{ cm}$$? (Recall: $$V = \frac{4}{3}\pi r^3$)

Accepted Answer

$$\frac{dr}{dt} = \frac{100}{4\pi (5^2$)} = \frac{1}{\pi}$$

Question 2

Given $$f(x) = x^2$ \sin(x)$$, which rule should be used to find $$f'(x)$$ and what is the correct derivative?

Accepted Answer

Product rule; $$f'(x) = 2x \sin(x) + x^2$ \cos(x)$$

Question 3

If $$y^2$$ + $$x^2 = 25$, and $$\frac{dx}{dt} = 3$$ when x = 4$ and y = 3$, what is $$\frac{dy}{dt}$$ at that instant?

Accepted Answer

$$\frac{dy}{dt} = -4$$

Question 4

Which of the following is the correct application of the quotient rule for $$f(x) = \frac{x^2$}{\sin(x)}$$?

Accepted Answer

$$f'(x) = \frac{2x \sin(x) - x^2$ \cos(x)}{\sin^2$(x)}$$

Question 5

A water tank in the shape of a cone is being filled at a rate of $$2 	ext{ ft}^3/	ext{min}. $$The tank has a height of $$10 	ext{ ft}$$ and a radius of $$4 	ext{ ft}. $$How fast is the water level rising when the water is $$6 	ext{ ft}$$ deep?

Accepted Answer

$$\frac{dh}{dt} = \frac{2}{\pi (16/100) (36)}$$

Question 6

Find the derivative of $$f(x) = \sin(x) \cos(x)$$ using the product rule.

Accepted Answer

$$f'(x) = \cos(x) \cos(x) + \sin(x) (-\sin(x)) = \cos^2$(x) - \sin^2$(x)$$

Question 7

Which of the following is a common mistake when applying the product rule?

Accepted Answer

Multiplying the derivatives: $$f'(x) = u'(x) v'(x)$$

Question 8

Given $$f(x) = (3x^2$ + 2x) e^x$$, find $$f'(x)$$ using the product rule.

Accepted Answer

$$f'(x) = (6x + 2) e^x + (3x^2$ + 2x) e^x$$

Question 9

If $$y^2 + xy = 10$, find $$\frac{dy}{dx}$$ using implicit differentiation.

Accepted Answer

$$\frac{dy}{dx} = \frac{-x}{2y + x}$$

Question 10

Which of the following is the correct derivative of $$f(x) = \ln(x^2$ + 1)$$ using the chain rule?

Accepted Answer

$$f'(x) = \frac{2x}{x^2$ + 1}$$

Question 11

A circular oil spill is expanding. If the radius increases at $$2 	ext{ ft/min}$$, how fast is the area increasing when $$r = 10 	ext{ ft}$$? (Recall: $$A = \pi r^2$)

Accepted Answer

$$\frac{dA}{dt} = 2\pi r \frac{dr}{dt} = 2\pi (10)(2) = 40\pi 	ext{ ft}^2/	ext{min}$$

Question 12

Which of the following is the correct derivative of $$f(x) = 	an(x)$$?

Accepted Answer

$$f'(x) = \sec^2$(x)$$

Question 13

If $$f(x) = x^3$ + 2x^2$ - 5$$, what is $$f'(x)$$ using the power rule and sum rule?

Accepted Answer

$$f'(x) = 3x^2$ + 4x$$

Question 14

Which of the following is a correct step when using implicit differentiation for $$x^2$$ + $$y^2 = 25$?

Accepted Answer

Differentiate both sides: $$2x + 2y \frac{dy}{dx} = 0$$

Question 15

A ladder $$10 	ext{ ft}$$ long is sliding down a wall. If the bottom slides away at $$2 	ext{ ft/min}$$, how fast is the top sliding down when the bottom is $$6 	ext{ ft}$$ from the wall?

Accepted Answer

$$\frac{dy}{dt} = -\frac{6}{8} 	imes 2 = -1.5 	ext{ ft/min}$$

Question 16

Which of the following is the correct derivative of $$f(x) = e$$^{2x}$$ using the chain rule?

Accepted Answer

$$f'(x) = 2e$$^{2x}$$

Question 17

Which of the following is NOT a recommended strategy for related rates problems?

Accepted Answer

Substitute known values before differentiating

Question 18

If $$f(x) = \sin(x^2$)$$, what is $$f'(x)$$ using the chain rule?

Accepted Answer

$$f'(x) = 2x \cos(x^2$)$$

Calculus I: Differentiation Rules, Applications, and Exam Strategies

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