{Use of Tech} Tangent lines Determine equations of the lines tangent to the graph of y= x√5−x² at the points (1, 2) and (−2,−2). Graph the function and the tangent lines.

Verified step by step guidance

1. Step 1: Find the derivative of the function y = x\sqrt{5-x^2}. Use the product rule and chain rule to differentiate y with respect to x.
2. Step 2: Evaluate the derivative at the point (1, 2) to find the slope of the tangent line at this point. Substitute x = 1 into the derivative to get the slope.
3. Step 3: Use the point-slope form of a line, y - y_1 = m(x - x_1), where m is the slope found in Step 2 and (x_1, y_1) is the point (1, 2), to write the equation of the tangent line at (1, 2).
4. Step 4: Repeat Steps 2 and 3 for the point (-2, -2). Evaluate the derivative at x = -2 to find the slope of the tangent line at this point, and use the point-slope form to write the equation of the tangent line.
5. Step 5: Graph the original function y = x\sqrt{5-x^2} and the two tangent lines found in Steps 3 and 4 on the same set of axes to visualize the function and its tangent lines at the given points.

Key Concepts

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line is equal to the derivative of the function at that point, which represents the instantaneous rate of change of the function. To find the equation of the tangent line, we use the point-slope form of a line, which requires the slope and the coordinates of the point of tangency.

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. For the function y = x√(5 - x²), the derivative can be calculated using the product and chain rules, which will provide the slope needed for the tangent lines at specific points.

Graphing Functions

Graphing a function involves plotting its output values against its input values on a coordinate plane. This visual representation helps in understanding the behavior of the function, including its intercepts, increasing/decreasing intervals, and points of tangency. For the given function, graphing it alongside the tangent lines allows for a clear visual comparison of how the tangent lines relate to the curve at the specified points.