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Semester Exam Review

Semester Exam Review. AP Calculus. Limits (algebraically & by graph) Find derivatives using limit definition Given a graph, sketch derivative graph Derivatives Power Rule Chain Rule Product/Quotient Rules , , ln x Trig derivatives Inverse trig derivatives

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Semester Exam Review

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  1. Semester Exam Review AP Calculus

  2. Limits (algebraically & by graph) • Find derivatives using limit definition • Given a graph, sketch derivative graph • Derivatives • Power Rule • Chain Rule • Product/Quotient Rules • , , ln x • Trig derivatives • Inverse trig derivatives • Indefinite Integrals (+ c !!!) • Definite Integrals (Fundamental Theorem) Exam Topics

  3. Trig function derivatives

  4. Co-Functions: Negative!! Differentiate:

  5. “forward difference quotient” Find f’(x) using the limit definition of derivatives:

  6. = = Derivatives by limits:

  7. Average Rate of Change – NOT AN AVERAGE!!! (slope of secant!): Units: mi/hr, ft/s, etc. Rate of Change:Average vs. Instantaneous

  8. DERIVATIVE!!!! (Slope of tangent line) Instantaneous Rate of Change =

  9. Write the equation of the tangent line to f(x) at x = 2 if -- Use slope-intercept form: y – y1 = m(x – x1) f(2) = 19, so (2, 19) is a point on graph -- Use derivative to find slope of tan. at x = 2. f’(x) = 6x  6(2) = 12 y – 19 = 12(x – 2) (can write in slope-int form as well) Writing Equation of Tangent Line

  10. Displacement function: Derivative of displacement is velocity: Derivative of velocity is acceleration: a(t) = 30t Displacement/Velocity/Acceleration

  11. Implicit Differentiation:

  12. Derivative: Differentiate implicitly:

  13. To the nearest thousandth, calculate the slope of the tangent where x = 4:

  14. Differentiate implicitly: Find coordinates of y when x = 4 and substitute into dy/dx equation: To the nearest thousandth, calculate the slope of the tangent where x = 4:

  15. Cylinder Volume: V = Useful Related Rates Formulas

  16. Be able to find values that make a piecewise function differentiable at a given point (must be continuous AND differentiable) Remember to use LIMITS to show differentiability and continuity! Differentiability Implies Continuity

  17. Function f is continuous at x = c if and only if: 1.) f(c) exists 2.) 3.) To Prove Continuity:

  18. Make sure to draw graph! • Check on calculator when possible, but show all setup • Remember that the AP exam tends to use uneven intervals so you have to do it by hand • Trapezoid area: Trapezoidal Rule/Riemann Sums

  19. Implicit Differentiation • Differential dy • Average Rate of Change • Instantaneous Rate of Change • Estimate definite integrals using trap rule, Riemann sums, graph • Applications • Find velocity given displacement equation • Find displacement given velocity equation • Write equation of tangent line • Find c value guaranteed by Mean Value Theorem • Related Rates • Calculator • Find numerical derivatives • Table of Values/Graph • Riemann Sums/Trapezoidal Rule Program (especially for large values of n!) Exam Topics Checklist

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