Related Rates and Derivatives

What is the limit of (1 - x^2) / (x^4 - x) as x approaches 1?

• -2

• -2/3

• 2/3

• 1

Evaluate the limit: lim (x→3) (x^2 - 9) / (x - 3)

• 0

• 3

• 6

• The limit does not exist.

What is the derivative of the function f(x) = ln(sin^2(x) + 3)?

• 2sin(x)cos(x) / (sin^2(x) + 3)

• 1 / (sin^2(x) + 3)

• cos(x) / (sin^2(x) + 3)

• 2sin(x) / (sin^2(x) + 3)

What is the slope of the line tangent to the graph of the function f(x) = ln(sin^2(x) + 3) at the point where x = pi/3?

• 1/15

• sqrt(3)/15

• 2sqrt(3)/15

• 4/15

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• Yes

• No

Given f(x) = sqrt(2x + 1), if g(x) is the inverse function of f(x), what is g'(3)?

• 1/(2sqrt(7))

• 1/sqrt(7)

• 1/3

• 3

Oil is poured on a flat surface, spreading out to form a circle. The area of this circle is increasing at a constant rate of 5 cm²/s. At what rate, in cm/s, is the radius of the circle increasing when the radius is 5 cm?

• 1/(2pi)

• 1/pi

• 1

• pi

When solving a related rates problem involving the area of a circle (A = πr²), what is the correct derivative of the area with respect to time (dA/dt)?

• dA/dt = π * dr/dt

• dA/dt = 2πr

• dA/dt = 2πr * dr/dt

• dA/dt = πr² * dr/dt

Oil is poured on a flat surface, spreading out to form a circle. If the area of the circle is increasing at a constant rate of 5 cm²/s, at what rate is the radius of the circle increasing when the radius is 5 cm?

• 1/(2π) cm/s

• 1/π cm/s

• 1 cm/s

• π cm/s

If f is a continuous function on [a, b] and M is the maximum value of f on [a, b], what is the upper bound for the definite integral of f(x) from a to b?

• ∫[a,b] f(x)dx ≤ M

• ∫[a,b] f(x)dx ≤ M * (b - a)

• ∫[a,b] f(x)dx ≤ (b - a) / M

• ∫[a,b] f(x)dx ≤ M / (b - a)