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An Introduction to Calculus

An Introduction to Calculus. Calculus. Study of how things change Allows us to model real-life situations very accurately. Topics (not in order). Limits Slope of a curve Tangent Lines Extreme values of a curve Area under a curve/volume of a region Lengths

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An Introduction to Calculus

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  1. An Introduction to Calculus

  2. Calculus • Study of how things change • Allows us to model real-life situations very accurately

  3. Topics (not in order) • Limits • Slope of a curve • Tangent Lines • Extreme values of a curve • Area under a curve/volume of a region • Lengths • Riemann Sums (Finite & Infinite) • Rates of Change

  4. Before CalculusAfter Calculus • Find slope of a line • Calculate average speed of a car • Calculate area/volume of rigid geometric figures • Find slope of a curve at a point • Determine exact speed of a car at a particular time • Determine area/volume of any figure under any curve/region

  5. Two Main Problems to Solve: I. Tangent Line Problem: How do you find the equation of the tangent line to the graph of a curve at a particular point? II. Area Under Curve Problem: How can you find the area underneath a curve and above the x-axis?

  6. Secant line f(c+Dx) Secant line f(c+Dx) Tangent line f(c) c c+Dx c+Dx

  7. As Dx decreases (Dx→0), secant line approaches the tangent line.

  8. Dx Dx Dx Dx a x1 x2 x3 b

  9. Dx Dx Dx Dx Dx Dx Dx Dx a x1 x2 x4 x5 x6 x7 b x3

  10. As the number of rectangles increases, the sum of their areas approaches the exact area of the curve.

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