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Math 1304 Calculus I

Math 1304 Calculus I. 2.1 – Tangent Lines and Velocity. What this chapter is about. Limits Limit as an extension to the idea of evaluating functions, giving it a natural value where it was not previously defined Limits are needed in the following situation Instantaneous rate of change

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Math 1304 Calculus I

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  1. Math 1304 Calculus I 2.1 – Tangent Lines and Velocity

  2. What this chapter is about • Limits • Limit as an extension to the idea of evaluating functions, giving it a natural value where it was not previously defined • Limits are needed in the following situation • Instantaneous rate of change • Average rate of change = ratio of change in function’s value to its change in argument • Instantaneous rate of change = limit of average rate of change as change gets smaller • Tangent lines and slope • Slope of curve is an example of instantaneous rate of change

  3. Review Straight Lines • Straight lines are often used to model more complicated curves • Some facts about straight lines • Simple geometry: Two distinct points determine a line • Simple equations: • General equation A x + B y + C = 0 • As the graph of a function y = m x + b, where m is the slope and b is the y-intercept • Point-slope formula (y – y0)/(x – x0) = m

  4. Tangent line to Curve • Problem: • Find the tangent line to a given curve at a given point Tangent Line Curve Point

  5. Tangent • “Tangent” comes from the Latin word tangens which means “touching”. • Given a curve and a point on that curve, a tangent line to the curve at that point is a line that has maximal contact with the curve at that point. Not only does it touch, but it goes in the same direction as the curve at that point. Tangent Line Curve Point

  6. Finding Tangent Lines Given a curve and a point on that curve, a second point on that curve determines a line called the secant line. 2nd pt 1st pt

  7. Tangent Lines as Limits As the second point gets closer to the first point, the secant line gets closer to the tangent line. Tangent Line 1st pt

  8. Tangent to graph of function • Problem: Given a curve y = f(x) and point x = x0 Find the tangent line to the curve at x = x0 • Solution: Find the slope of the curve y = f(x) at x = x0 and use the point slope formula for a line

  9. Slope of Graph of a Function y0=f(x0) 1st pt x0

  10. 2nd pt y = f(x) y0=f(x0) 1st pt x0 x Slope of Graph of a Function

  11. 2nd pt y=f(x) y0=f(x0) 1st pt x0 x Slopes of Graph of a Function

  12. 2nd pt y=f(x) y0=f(x0) 1st pt x0 x Slopes of Graph of a Function

  13. Slopes of Graph of a Function y=f(x0) 1st pt x0

  14. "Smooth" curves have tangents • Intuitively, smooth means no corners.

  15. Examples • Work out in class: f(x) = x2, with x0=1 f(x) = x2, with x0=2 • Method • Find y0=f(x0) • Evaluate (f(x)-f(x0 ))/(x-x0 ) • Take limit as x gets closer to x0

  16. 2nd pt f(x+h) f(x) 1st pt x x+h Slopes using increments h

  17. Examples • Work out in class: f(x) = x2 f(x) = x3

  18. Tangent lines to graphs of functions • Now that we can do slopes, what about tangents? • Procedure: • For a given function at a given point find the slope • Plug it into the slope-point formula • Example in class: f(x)=x2 at x = 2

  19. Velocity • Velocity is the rate of change of position.

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