Exploring Calculus Concepts

What is the derivative of sin(x)?

• tan(x)

• sec(x)

• sin^2(x)

• cos(x)

Calculate the integral of x^2 from 0 to 3.

• 6

• 9

• 15

• 12

What is the limit of (1/x) as x approaches infinity?

• 0

• infinity

• -1

• 1

Find the second derivative of f(x) = 3x^3 - 5x + 2.

• f''(x) = 9x^2 - 5

• f''(x) = 18x

• f''(x) = 36x^2

• f''(x) = 6x^2

Evaluate the integral of e^x dx.

• e^x + 1

• e^x + C

• e^x - C

• e^x / x

What is the fundamental theorem of calculus?

• The fundamental theorem of calculus states that all continuous functions are differentiable.

• The fundamental theorem of calculus states that the area under a curve can be found using limits.

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

• The fundamental theorem of calculus states that the derivative of a function is equal to its integral.

Determine the critical points of f(x) = x^4 - 4x^2.

• x = 2

• x = 0, x = sqrt(2), x = -sqrt(2)

• x = 1

• x = -1

What is the derivative of ln(x)?

• ln(1/x)

• x

• 1

• 1/x

Calculate the area under the curve y = x^3 from x = 1 to x = 2.

• 3.75

• 5.0

• 4.0

• 2.5

What is the relationship between a function and its derivative?

• The derivative indicates the maximum value of a function.

• The derivative of a function measures its rate of change.

• The derivative shows the function's domain.

• The derivative of a function is its integral.