Understanding Tangent Lines and Derivatives

The video tutorial explains how to find the equation of a tangent line to the function f(x) = x e^x at x = 1. It begins with an introduction to the problem, followed by graphing the function. The product rule is used to find the derivative, which is then evaluated at x = 1 to determine the slope of the tangent line. The equation of the tangent line is derived using the slope and a known point. Finally, the solution is verified using a graphing calculator.

What is the main objective of the problem discussed in the video?

• To calculate the integral of the function

• To determine the equation of the tangent line

• To find the area under the curve

• To solve a system of equations

What is the function f(x) that is being analyzed in the video?

• f(x) = x^2

• f(x) = e^x

• f(x) = x * e^x

• f(x) = x + e^x

Which mathematical rule is used to find the derivative of the function?

• Chain Rule

• Product Rule

• Power Rule

• Quotient Rule

What is the slope of the tangent line at the point x = 1?

• e

• 2e

• 3e

• 4e

What is the y-intercept of the tangent line equation derived in the video?

• e

• -e

• 0

• 2e

What is the final equation of the tangent line?

• y = ex - e

• y = 2ex + e

• y = 2ex - e

• y = ex + 2e

How is the accuracy of the tangent line equation verified?

• By comparing with another function

• By using a graphing calculator

• By solving algebraically

• By checking with a table of values

What is the significance of the number 'e' in the context of this problem?

• It is a variable

• It is a constant like pi

• It is the slope of the tangent line

• It is the y-intercept

What is the value of f(1) for the function f(x) = x * e^x?

• 0

• e

• 1

• 2e

What is the purpose of using the product rule in this problem?

• To solve for the y-intercept

• To determine the slope of the tangent line

• To calculate the area under the curve

• To find the integral of the function