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Derivative as a Rate of Change

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  1. Derivative as a Rate of Change Chapter 3 Section 4

  2. Usually omit instantaneous Interpretation: The rate of change at which f is changing at the point x Interpretation: Instantaneous rate are limits of average rates.

  3. Example • The area A of a circle is related to its diameter by the equation • How fast does the area change with respect to the diameter when the diameter is 10 meters? • The rate of change of the area with respect to the diameter • Thus, when D = 10 meters the area is changing with respect to the diameter at the rate of

  4. Motion Along a Line • Displacement of object over time Δs = f(t + Δt) – f(t) • Average velocity of object over time interval

  5. Velocity • Find the body’s velocity at the exact instant t • How fast an object is moving along a horizontal line • Direction of motion (increasing >0 decreasing <0)

  6. Speed • Rate of progress regardless of direction

  7. Graph of velocity f ’(t)

  8. Acceleration • The rate at which a body’s velocity changes • How quickly the body picks up or loses speed • A sudden change in acceleration is called jerk • Abrupt changes in acceleration

  9. Example 1: Galileo Free Fall • Galileo’s Free Fall Equation s distance fallen g is acceleration due to Earth’s gravity (appx: 32 ft/sec2 or 9.8 m/sec2) • Same constant acceleration • No jerk

  10. Example 2: Free Fall Example • How many meters does the ball fall in the first 2 seconds? • Free Fall equation s = 4.9t2 in meters s(2) – s(0) = 4.9(2)2 - 4.9(0)2 = 19.6 m

  11. Example 2: Free Fall Example • What is its velocity, speed and acceleration when t = 2? • Velocity = derivative of position at any time t • So at time t = 2, the velocity is

  12. Example 2: Free Fall Example • What is its velocity, speed and acceleration when t = 2? • Velocity = derivative of position at any time t • So at time t = 2, the speedis

  13. Example 2: Free Fall Example • What is its velocity, speed and acceleration when t = 2? • Velocity = derivative of position at any time t • The acceleration at any time t • So at t = 2, acceleration is (no air resistance)

  14. Derivatives of Trigonometric Functions Chapter 3 Section 5

  15. Derivatives

  16. Application: Simple Harmonic Motion • Motion of an object/weight bobbing freely up and down with no resistance on an end of a spring • Periodic, repeats motion • A weight hanging from a spring is stretched down 5 units beyond its rest position and released at time t = 0 to bob up and down. Its position at any later time t is s = 5 cos(t) • What are its velocity and acceleration at time t?

  17. Application: Simple Harmonic Motion • Its position at any later time t is s = 5 cos(t) • Amp = 5 • Period = 2 • What are its velocity and acceleration at time t? • Position: s = 5cos(t) • Velocity: s’ = -5sin(t) • Speed of weight is 0, when t = 0 • Acceleration: s’’ = -5 cos(t) • Opposite of position value, gravity pulls down, spring pulls up

  18. Chain Rule Chapter 6 Section 6

  19. Implicit Differentiation Chapter 3 Section 7

  20. Implicit Differentiation • So far our functions have been y = f(x) in one variable such as y = x2 + 3 • This is explicit differentiation • Other types of functions x2+ y2 = 25 or y2 – x = 0 • Implicit relation between the variables x and y • Implicit Differentiation • Differentiate both sides of the equation with respect to x, treating y as a differentiable function of x (always put dy/dx after derive y term) • Collect the terms with dy/dx on one side of the equation and solve for dy/dx

  21. Circle Example

  22. Folium of Descartes • The curve was first proposed by Descartes in 1638. Its claim to fame lies in an incident in the development of calculus. • Descartes challenged Fermat to find the tangent line to the curve at an arbitrary point since Fermat had recently discovered a method for finding tangent lines. • Fermat solved the problem easily, something Descartes was unable to do. • Since the invention of calculus, the slope of the tangent line can be found easily using implicit differentiation.

  23. Folium of Descartes • Find the slope of the folium of Descartes • Show that the points (2,4) and (4,2) lie on the curve and find their slopes and tangent line to curve

  24. Folium of Descartes • Show that the points (2,4) and (4,2) lie on the curve and find their slopes and tangent line to curve

  25. Folium of Descartes • Show that the points (2,4) and (4,2) lie on the curve and find their slopes and tangent line to curve • Find slope of curve by implicit differentiation by finding dy/dx

  26. PRODUCT RULE

  27. Factor out dy/dx Divide out 3

  28. Evaluate at (2,4) and (4,2) Slope at the point (2,4) Slope at the point (4,2)

  29. Find Tangents

  30. Folium of Descartes • Can you find the slope of the folium of Descartes • At what point other than the origin does the folium have a horizontal tangent? • Can you find this?

  31. Derivatives of Inverse Functions and Logarithms Chapter 3 Section 8

  32. Examples 1 & 2