Deriv basics (pt 2 )
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The document discusses key concepts in calculus including: - The derivative of a function gives the instantaneous rate of change or slope of a curve at a given point. - The secant line represents average rate of change, while the tangent line at a point represents instantaneous rate of change. - Differentiation is the process of finding the derivative of a function. A differentiable function has a derivative. - An example problem asks to estimate the instantaneous rate of change of temperature given a temperature function and table of values.
Deriv basics (pt 2 )
- 1. AP Calculus Warm up 9.14.18 What is the derivative of a function? The derivative of a function is also a function which gives us the instantaneous rate of change (or slope) of the curve at any given point.
- 2. The Lame Joke of the day.. And now it’s time for.. What did one shooting star say to the other? Pleased to meteor.
- 3. AP Calculus Warm up 9.14.18 What is the derivative of a function? The derivative of a function is also a function which gives us the instantaneous rate of change (or slope) of the curve at any given point.
- 4. Secant line = Average rate of change
- 5. The secant line becomes the tangent line
- 7. Tangent line = instantaneous rate of change.
- 8. Rates of change Average rate of change slope of secant line Use slope formula Instantaneous rate of change slope of Tangent line Use derivative
- 10. Differentiation • The process of finding the derivative of a function • If a function has a derivative, it is called “differentiable”
- 11. Find the Instantaneous Rate of change at (2, 12) 2 3)( xxf
- 12. What is the equation of the tangent line of f(x) at ( 2, 12) 2 3)( xxf Slope = 12
- 13. Time (t) minutes 0 2 9 20 35 Temperature C (t) in degrees Celsius 60 63 89 102 110 A beaker is being heated continuously over time (t) and is modeled by the twice differentiable function C (t). The table above gives selected values of the temperature in degrees Celsius. Estimate the instantaneous rate of change of the temperature at 14 minutes. Indicate units of measure.
- 14. What makes a function differentiable? 1. It MUST be continuous. 2. The derivative from the left must equal the derivative from the right.
- 15. Most common times when a derivative does not exist. 1. At a sharp corner 2. At a vertical tangent line or asymptote
- 16. KEY POINT • Differentiability implies continuity, But continuity DOES NOT imply differentiability.
- 17. Notation
- 18. Review The derivative of a function is also a function which gives us the instantaneous rate of change (or slope) of the curve at any given point.
- 19. The graph of f is given below. Sketch out the graph of f ’