Determine the equation of the tangent to the curve defined by y=x e^-x at the point A(1, e^-1).

Name: Problem 7, Chapter 5: Nelson Calculus and Vectors
Uploaded: 2021-12-12T23:34:02-08:00
Duration: 1 min 46 s
Channel: Gregory Higby
Description: Determine the equation of the tangent to the curve defined by y=x e^-x at the point A(1, e^-1).

step 1 So with this problem, they give you a lot of information and it's also easy to mess up. But the

Determine the equation of the tangent to the curve defined...

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Key Concepts

Tangent Line

A tangent line to a curve at a given point is the straight line that touches the curve only at that point and has the same instantaneous rate of change (slope) as the curve at that point. It serves as the best linear approximation of the curve near the point of tangency.

Product Rule

The product rule is a fundamental differentiation technique used when differentiating a product of two functions. It states that if you have a function y = f(x) * g(x), its derivative y' is given by f'(x) * g(x) + f(x) * g'(x), ensuring that both functions' rates of change are accounted for.

Differentiation of Exponential Functions

Exponential functions, such as those involving the constant e, have the property that their derivative is proportional to the function itself. This characteristic makes them crucial in problems involving growth and decay, and understanding their differentiation is essential in calculus.

Point-Slope Form

The point-slope form is a method for expressing the equation of a straight line when a point on the line and the slope are known. It is typically written as y - y? = m(x - x?), providing an easy and direct way to obtain the linear equation once the slope from the derivative is determined.

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