Cold Calculus

What is the alternate definition of the derivative f'(c)?

• f'(c) = lim (f(x) + f(c)) / (x + c)

• f'(c) = lim (f(x) - f(c)) / (x - c)

• f'(c) = lim (f(x) * f(c)) / (x * c)

• f'(c) = lim (f(x) / f(c)) * (x / c)

According to the Intermediate Value Theorem, if the function f(x) is continuous on [a, b] and y is a number between f(a) and f(b), what is guaranteed?

• There exists at least one number x = c in (a, b) such that f'(c) = y.

• There exists at least one number x = c in (a, b) such that f(c) = y.

• There exists at least one number x = c in (a, b) such that f''(c) = y.

• There exists at least one number x = c in (a, b) such that f(c) = 0.

What does the Mean Value Theorem state about a function f(x) that is continuous on [a, b] and has a first derivative that exists on the interval (a, b)?

• There exists at least one number x = c in (a, b) such that f(c) = f(b) - f(a) / b - a.

• There exists at least one number x = c in (a, b) such that f''(c) = f(b) - f(a) / b - a.

• There exists at least one number x = c in (a, b) such that f'(c) = f(b) - f(a) / b - a.

• There exists at least one number x = c in (a, b) such that f'(c) = 0.

Rolle's Theorem applies to a function f(x) that is continuous on [a, b], has a first derivative that exists on (a, b), and satisfies which additional condition?

• f(a) = f(b) and f'(c) = 0 for some c in (a, b).

• f(a) = f(b) and f(c) = 0 for some c in (a, b).

• f(a) ≠ f(b) and f'(c) = 0 for some c in (a, b).

• f(a) ≠ f(b) and f(c) = 0 for some c in (a, b).

What does the Extreme Value Theorem guarantee for a function f(x) that is continuous on [a, b]?

• The function has at least one x = c in (a, b) where f(c) is a maximum or minimum.

• The function has at least one x = c in (a, b) where f'(c) is a maximum or minimum.

• The function has an absolute maximum and an absolute minimum on the interval (a, b).

• The function has an absolute maximum and an absolute minimum on the interval [a, b].

What is the derivative of cos(x) with respect to x?

• sin(x)

• -sin(x)

• sec^2(x)

• -csc^2(x)

According to the Chain Rule, what is the derivative of f(g(x)) with respect to x?

• f'(g(x)) * g'(x)

• f'(x) * g'(x)

• f'(g(x)) / g'(x)

• f'(x) / g'(x)

What is the Product Rule for finding the derivative of the product of two functions u(x) and v(x)?

• (u * v)' = u' * v + u * v'

• (u * v)' = u' / v + u / v'

• (u * v)' = u' * v - u * v'

• (u * v)' = u' / v - u / v'

What is the Quotient Rule for finding the derivative of the quotient of two functions u(x) and v(x)?

• (u / v)' = (u' * v - u * v') / v^2

• (u / v)' = (u' * v + u * v') / v^2

• (u / v)' = (u' / v - u / v') / v^2

• (u / v)' = (u' / v + u / v') / v^2

What is the derivative of an inverse function g(x) if f has an inverse function g?

• g'(x) = f'(g(x))

• g'(x) = 1 / f'(g(x))

• g'(x) = f(g(x))

• g'(x) = -1 / f'(g(x))