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Find the slope of the tangent to the curve y = 3 + 4x² - 2x³ at the point where x = a A. THE DERIVATIVE AS A SLOPE Recall: From Analytic Geometry, the equation of a line (slope intercept form) is y=

Find the slope of the tangent to the curve y = 3 + 4x² - 2x³ at the point where x = a A. THE DERIVATIVE AS A SLOPE Recall: From Analytic Geometry, the equation of a line (slope intercept form) is y=

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 92E
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2. Find the slope of the tangent to the curve y = 3 + 4x² - 2x³ at the point where x = a A. THE DERIVATIVE AS A SLOPE Recall: From Analytic Geometry, the equation of a line (slope intercept form) is y= mx + b, where m is the slope of the line. Following the determination of derivatives, SAPIE y + Ay = m(x + Ax) + b Ay = m(x + Ax) + b-mx-b Ay = max y=2=x² Ay dy Ax dx = Illustrative Example: Find the slope of the curve y = 2x² at the point (3,-7). To determine the slope of the tangent line to the curve, we get the derivative of the curve y = 2-x² Tangent line OP(3,-7) is the slope of the tangent line at a certain or particular point on the curve y+Ay=2-(x + Ax)² Ay = 2(x +Ax)² - y Ay= 2(x+Ax)²-2+x² Ay=2-x²2xAx-Ax²-2+x² Ay ==-2x-Ax Ax Ay lim = -2x This is the slope of the Ax-0 Ax dy dx Therefore, the slope at P(3.-7) or at x = 3 is -=-2 (3) = -6 tangent line to the curve at any given point
Transcribed Image Text:2. Find the slope of the tangent to the curve y = 3 + 4x² - 2x³ at the point where x = a A. THE DERIVATIVE AS A SLOPE Recall: From Analytic Geometry, the equation of a line (slope intercept form) is y= mx + b, where m is the slope of the line. Following the determination of derivatives, SAPIE y + Ay = m(x + Ax) + b Ay = m(x + Ax) + b-mx-b Ay = max y=2=x² Ay dy Ax dx = Illustrative Example: Find the slope of the curve y = 2x² at the point (3,-7). To determine the slope of the tangent line to the curve, we get the derivative of the curve y = 2-x² Tangent line OP(3,-7) is the slope of the tangent line at a certain or particular point on the curve y+Ay=2-(x + Ax)² Ay = 2(x +Ax)² - y Ay= 2(x+Ax)²-2+x² Ay=2-x²2xAx-Ax²-2+x² Ay ==-2x-Ax Ax Ay lim = -2x This is the slope of the Ax-0 Ax dy dx Therefore, the slope at P(3.-7) or at x = 3 is -=-2 (3) = -6 tangent line to the curve at any given point
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