Understanding Tangent and Normal Lines

The video tutorial explains how to find the equations of the tangent and normal lines to the function f(x) = x^3 + 2e^x at the point (0, 2). It covers the calculation of the derivative to determine the slope of the tangent line, and then uses the negative reciprocal to find the slope of the normal line. The tutorial also includes a graphical verification of the results.

What is the function given in the problem?

• f(x) = x^2 - 2e^x

• f(x) = x^3 - 2e^x

• f(x) = x^3 + 2e^x

• f(x) = x^2 + 2e^x

What is the first step in finding the equation of the tangent line?

• Evaluate the function at the point

• Calculate the derivative

• Determine the normal line

• Find the y-intercept

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

• 3x^2 - 2e^x

• x^2 + 2e^x

• 3x^2 + 2e^x

• x^3 + 2e^x

What is the slope of the tangent line at the point (0, 2)?

• -1

• 0

• 2

• 1

What is the equation of the tangent line at the point (0, 2)?

• y = 2x + 2

• y = x + 2

• y = x - 2

• y = 2x - 2

What is the relationship between the slopes of the tangent and normal lines?

• They are equal

• They are negative reciprocals

• They are both positive

• They are both zero

What is the slope of the normal line at the point (0, 2)?

• 2

• -1/2

• 1/2

• -2

What is the equation of the normal line at the point (0, 2)?

• y = -1/2x + 2

• y = 1/2x + 2

• y = 2x - 2

• y = -2x + 2

How is the correctness of the tangent and normal lines verified?

• By checking with a calculator

• By solving algebraically

• By comparing with another function

• By graphing the lines

What color is used to represent the function f(x) in the graph?

• Green

• Red

• Yellow

• Blue