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- 1. { The Derivative Slope of a line, secant line, tangent line, slope of a secant line, slope of a line tangent to a curve
- 2. Lines, slope, point-slope form
- 3. What do we know about slopes of lines? Lines, slope, point-slope form
- 4. Lines, slope, point-slope form
- 5. Lines, slope, point-slope form
- 6. Lines, slope, point-slope form
- 7. Given a curve, the line that intersects the curve on two points is called a secant line. Secant line
- 8. Given a curve, the line that intersects the curve on two points is called a secant line. Secant line
- 9. Tangent line
- 10. Given a non-linear curve, how do you get the slope of the curve at a the point (x, f(x) )? Challenge question
- 11. Given a non-linear curve, how do you get the slope of the curve at a the point (x, f(x) )? Challenge question Let 𝑚 𝑇𝐿 be the slope of the tangent line, then the slope of the curve at (x,f(x)) is the same as 𝑚 𝑇𝐿 𝑚 𝑇𝐿 is the derivative of f at x. 𝑓′ 𝑥 = 𝑚 𝑇𝐿 𝑦 = 𝑓(𝑥)
- 12. How do we compute 𝑚 𝑇𝐿? Challenge question
- 13. How to find 𝑚 𝑇𝐿 𝑥𝑖, 𝑓 𝑥𝑖 𝑚 𝑇𝐿 = Δ𝑦 Δ𝑥 = 𝑦2 − 𝑦1 𝑥2 − 𝑥2 𝑓′ 𝑥 = lim Δ𝑥→0 𝑓 𝑥 + ∆𝑥 − 𝑓 𝑥 ∆𝑥
- 14. Three step rule 1. Find 𝑓 𝑥 + ∆𝑥 − 𝑓 𝑥 2. Find 𝑓 𝑥+∆𝑥 −𝑓 𝑥 ∆𝑥 1. Evaluate How to compute for 𝑓′ 𝑥 lim Δ𝑥→0 𝑓 𝑥 + ∆𝑥 − 𝑓 𝑥 ∆𝑥
- 15. How to compute for 𝑓′(𝑥) First step
- 16. How to compute for 𝑓′(𝑥) Second step
- 17. How to compute for 𝑓′(𝑥) Third step 𝑓′ 𝑥 = 4𝑥 − 3
- 18. What is 𝑓′ 2 ? 𝑓′ 2 = 4 2 − 3 = 5 What is 𝑓′ 0 ? 𝑓′ 0 = 4 0 − 3 = −3

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