Find an equation for the tangent line to the curve at the given point. Then sketch the curve and

Find an equation for the tangent line to the curve at the...

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Key Concepts

Point-Slope Form for Tangent Lines

Once the derivative (or slope) at the point of tangency is known, the point-slope form is used to write the equation of the tangent line. This form directly incorporates the slope and coordinates of the given point, making it a practical tool in analytic geometry.

Curve Sketching

Curve sketching combines the understanding of a function’s equation and its derivative information to create a visual representation. Plotting the function alongside its tangent line helps illustrate the concept of local linearity and provides insights into the behavior of the function near the point of tangency.

Differentiation

Differentiation involves finding the derivative of a function, which quantifies how the function’s output changes in response to changes in the input. It is the foundational tool in calculus used to compute rates of change.

Derivative as a Slope

The derivative of a function at a particular point gives the slope of the tangent line to the graph at that point. This concept is vital in understanding instantaneous rates of change in a variety of contexts.

Transcript

00:01 In this problem, we're asked to find the tangent line to the curve y equals x minus 1 squared plus 1 at the point 1 comma 1, and then to sketch the curve and the tangent line.

00:12 So to find the tangent line, we need to find the slope of the tangent line.

00:16 So the slope of the graph, slope of graph, at x equals 1.

00:24 So this is the x coordinate here.

00:28 So this is the limit expression from the section.

00:31 This is the limit as h approaches 0 of 1 plus h minus 1 squared plus 1.

00:41 That's our first term.

00:44 And then minus 1 minus 1 squared plus 1, all divided by h.

00:53 So where did this come from? so this x equals 1, that's coming right here and right here.

01:02 And so this part of the expression, this is the like x plus h part of the expression.

01:08 That's where this is one plus h is coming from.

01:10 Here's the x plus h.

01:11 Here's the x.