Determine equations of the lines tangent to

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The discussion focuses on determining the equations of the tangent lines to the graph of the function Y = x√(5-x²) at the points (1,2) and (-2,-2). To find the slope of the tangent line at these points, one must first compute the derivative of the function. The derivative provides the slope, which can then be used in the point-slope formula to derive the equations of the tangent lines. This process is essential for understanding how to analyze functions in calculus.

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  • Familiarity with the point-slope formula for linear equations
  • Basic knowledge of graphing functions and interpreting their behavior
  • Ability to compute square roots and handle algebraic expressions
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  • Learn how to compute derivatives using the product rule
  • Study the point-slope form of a linear equation in detail
  • Practice graphing functions and their tangent lines for better visualization
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Students taking calculus, particularly those struggling with derivatives and tangent lines, as well as educators seeking to provide clear explanations of these concepts.

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The problem is:
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.

I have no IDEA where to go with this. I am taking calculus over the summer and we are in week 2 and I'm struggling..if anyone could do a step by step process here and explain I would be so grateful. Thanks in advance.
 
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One of the first things you should have learned in Calculus is that the derivative of a function, at a given value of x, is the slope of the tangent line to the graph at that point on the graph. To find the slope of the tangent line at (2, 2) , find the derivative of [itex]y= x\sqrt{5- x^2}[/itex] at x= 2, then use the "point-slope" formula for the equation of the line having that slope through the line (2, 2).
 

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