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3.1. Tangent Lines and Rates of Change. Average and instantenious velocity.

Learn about tangent lines in calculus, slope calculations, and velocity concepts. Explore average and instantaneous rates of change, with examples and explanations.

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3.1. Tangent Lines and Rates of Change. Average and instantenious velocity.

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  1. 3.1.Tangent Lines and Rates of Change.Average and instantenious velocity. Rita Korsunsky

  2. . Tangent Lines Yes! • The tangent line to the graph of f is a line that touches the graph at one isolated point and could possibly intersect it again at another point. Is it a tangent to the curve at c? c Yes! . Is it a tangent to the curve at c? • Slope of the graph of f at point c is the slope of the tangent line at point c • . c not a tangent, 2 pts of intersection.

  3. Let’s first find the Slope of secant line: y . Q (x, f(x)) . P (c, f(c)) O x Slope of secant line =

  4. Finding the slope of the tangent line at pt C Let’s pick 2nd point closer and closer to C and calculate the slopes of Secant Lines. Watch the animation: Slope of tangent line at pt C  Slope of Secant Line when x is approaching to C

  5. Slope of tangent line at pt C  Slope of Secant Line when x is approaching to C Slope of tangent line = Let x - c = h x = c + h

  6. 1.Find the slope of tangentat any point x 2. Plug in x = c into mxto find the slope mc at the point (c, f(c)). To Find the Equation of the Tangent Line at x=c: 3. Substitute coordinates (c,f(c)) and slope mc into the point-slope equation of a line:

  7. Example 1 Solution:

  8. Average Velocity and rate of change

  9. Instantaneous Velocity and rate of change

  10. A sandbag is dropped from a hot-air balloon that is hovering at a height of 512 feet above the ground. If air resistance is disregarded, then the distance s(t) from the ground to the sand bag after t second is given by: Example 2 Find the velocity of the sandbag at: (a) t = a sec (b) t = 2 sec (c) the instant it strikes the ground (a) Find Velocity at t = a

  11. Example 2 continued (b) Find velocity at t = 2 sec (c) Find velocity at instant it hits the ground

  12. THE END

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