Exploring Calculus Concepts

What is the limit of f(x) as x approaches 2 for f(x) = 3x + 1?

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Define continuity at a point in terms of limits.

• A function f is continuous at a point c if f(c) is greater than lim (x -> c) f(x).

• A function f is continuous at a point c if lim (x -> c) f(x) is less than f(c).

• A function f is continuous at a point c if lim (x -> c) f(x) does not exist.

• A function f is continuous at a point c if lim (x -> c) f(x) = f(c).

What is the derivative of f(x) = x^3 - 5x + 4?

• x^2 - 5x + 4

• 3x^2 - 5

• 2x^2 - 5

• 3x^2 + 5

Explain the concept of the product rule in differentiation.

• The product rule is used to find the integral of two functions.

• The product rule states that (u*v)' = u' + v'.

• The product rule states that if u and v are functions, then (u*v)' = u'v + uv'.

• The product rule states that (u*v)' = uv' + u'v.

Calculate the integral of f(x) = 2x from x = 1 to x = 3.

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What is the Fundamental Theorem of Calculus?

• The Fundamental Theorem of Calculus states that the derivative of a function is equal to its integral.

• The Fundamental Theorem of Calculus states that the area under a curve can be found using limits only.

• The Fundamental Theorem of Calculus states that all continuous functions are differentiable.

• The Fundamental Theorem of Calculus states that if F is an antiderivative of f on [a, b], then ∫_a^b f(x) dx = F(b) - F(a).

Solve the differential equation dy/dx = 3y.

• y = Csin(3x)

• y = 3x + C

• y = Ce^(x)

• y = Ce^(3x)

What is the nth term of the sequence defined by a_n = 2n + 1?

• 3n + 2

• n^2 + 1

• 2n + 1

• 2n - 1

Determine the convergence or divergence of the series Σ (1/n^2) from n=1 to ∞.

• Oscillates

• Diverges

• Converges

• Converges conditionally

Explain the difference between definite and indefinite integrals.

• Definite integrals yield a number; indefinite integrals yield a function.

• Definite integrals represent area under a curve; indefinite integrals represent the slope of a curve.

• Definite integrals are always negative; indefinite integrals are always positive.

• Indefinite integrals require limits of integration; definite integrals do not.