Understanding Indefinite Integrals and Graph Shifts

The video tutorial explains how to find the general indefinite integral of 2x DX, focusing on the concept of anti-derivatives. It demonstrates the calculation process, resulting in x^2 + C, and discusses the graphical representation of this family of functions. The tutorial highlights how adding different constants shifts the graph vertically, illustrating the concept with examples of positive and negative constants.

What is the indefinite integral of 2x with respect to x?

• 2x^2 + C

• 2x + C

• x^3 + C

• x^2 + C

When finding the anti-derivative of 2x, what is the first step?

• Multiply by 2

• Increment the exponent by 1

• Add a constant

• Divide by 2

Why can we add a constant C to the anti-derivative?

• It modifies the curvature of the graph

• It alters the x-intercepts

• It accounts for all possible vertical shifts

• It changes the slope of the graph

What does the graph of y = x^2 + C represent?

• A single parabola

• A family of parabolas shifted horizontally

• A family of parabolas shifted vertically

• A line with varying slopes

If C is positive, how does the graph of y = x^2 + C shift?

• It shifts to the left

• It shifts downward

• It shifts upward

• It remains unchanged

What happens to the graph of y = x^2 + C if C is negative?

• It shifts upward

• It shifts downward

• It shifts to the right

• It becomes a line

Which of the following is true about the indefinite integral of 2x?

• It represents a family of functions

• It is a single function

• It is always negative

• It is always positive

What is the derivative of the function y = x^2 + C?

• 2x

• 0

• x^2

• C

How does the constant C affect the graph of y = x^2 + C?

• It shifts the graph vertically

• It shifts the graph horizontally

• It changes the shape of the parabola

• It rotates the graph

What is the general form of the indefinite integral of 2x?

• x^2 + C

• 2x + C

• x^3 + C

• 2x^2 + C