2.7: Tangents, Velocities, and Rates of ChangeWe are now ready to find a formal way of computing the slope of a line tangent to a curve at a point.Re-view the animation from 2.1 posted on my webpage. What happens as the second x coordinatemoves closer to the given tangent-line point?Each secant line above passes through the point (1, 1). If x is the x-coordinate of the second point,write an expression for the slope of the line between the two points.Write and solve a limit problem which allows us to find the slope of the tangent line at x = 1.More General: Draw any function, a tangent line, and a secant line in the space below. Label thetangent-line point (a, f(a)) and the second point on the secant line (x, f (x)).msec=What should happen as the point (x, f (x)) movescloser t o the ta ngent-line point (a, f(a))? Write alimit which explains this mathematically:mtan=Most General: Draw any function, a ta ng ent line, and a secant line again in the space below. Labelthe tangent-line point (a, f(a)) a nd let h be the distance between the x values on the secant line.View the new animation posted under t oday’s notes for a visual understanding of this).msec=mtan=The derivative of a function at x = a is given byExamples: Use a limit definition to find the equation of the line tangent to the curve f(x) =x2− 4x + 4 at the point where x = 3.Use a limit definition t o find the derivative of the function f(x) =1x − 2. Compute the slopes of thelines ta ngent to this graph at x = 0, x = 1, and x = 3.Secant and Tangent Vectors-an IntroductionIllustration of Velocity Vectors ( Secant and Tangent):Example: Find a vector tangent to the curve r(t) = (3t2)i +√t j at the p oint (3,1). Then findparametric equations of the line tangent to the curve at this
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