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PPT COT 3rd Demo for tangent lines .pptx
- 2. • For 10seconds, copy exactly what you can read on this slide. This is worth 5 points.
- 3. • How manyletter F’s are in this sentence? You have 10 seconds to count them. • “FINISHED FILES ARE THE RESULT OF YEARS OF SCIENTIFIC STUDY COMBINED WITH THE EXPERIENCE OF FORTUNATE YEARS.”
- 4. • For 10seconds, copy exactly what you can read on this slide. This is worth 5 points.
- 5. This time letus watch a video presentation about how can a single line affects our perspective.
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- 7. • Given afunction y=f(x), how do we find the equation of the tangent line at a point? • Consider the graph of a function y = f(x) whose graph is given below. Let P(x0, y0) be a point on the graph of y = f(x). • Our objective is to find the equation of the tangent line (TL) to the graph at the point P(x0, y0).
- 9. • Since thetangent line is the limiting position of the secant lines as Q approaches P, it follows that the slope of the tangent line (TL) at the point P is the limit of the slopes of the secant lines PQ as x approaches x0. In symbols, 𝑚𝑇𝐿 = lim 𝑥→𝑥0 𝑦 − 𝑦0 𝑥 − 𝑥0 = lim 𝑥→𝑥0 𝑓(𝑥) − 𝑓(𝑥0) 𝑥 − 𝑥0
- 10. • Finally, sincethe tangent line passes through P(x0, y0), then its equation is given by: 𝑦 − 𝑦0 = 𝑚𝑇𝐿(𝑥 − 𝑥0)
- 12. • EXAMPLE 1:Find the equation of the tangent line to y = x2 at x=2.
- 13. • Example 2.Find the slope-intercept form of the tangent line to
- 14. • Determine theEquation of the tangent line at the curve: • Group 1: y = x2 − 5x + 2 at (3,-4). • Group 2: y = x2 + 3x − 1 at x = 0. • Group 3: y = x2 − 2 at x = 0. • Group 4: y = 2x2 – 4x + 5 at x = -1 • Group 5: y = 3x2 – 12x + 1 at x = 0
- 15. Evaluation • Determine theEquation of the tangent line at the curve: • 𝒚 = 𝒙 + 𝟗 𝒂𝒕 𝒕𝒉𝒆 𝒑𝒐𝒊𝒏𝒕 𝒘𝒉𝒆𝒓𝒆 𝒙 = 𝟎. • 𝒚 = 𝟐𝟓 − 𝒙𝟐𝒂𝒕 𝒕𝒉𝒆 𝒑𝒐𝒊𝒏𝒕 𝒘𝒉𝒆𝒓𝒆 𝒙 = 𝟒.
- 16. Homework • Determine theEquation of the tangent line at the curve: • 𝒚 = 𝒙𝟐 + 𝒙 𝒂𝒕 𝒕𝒉𝒆 𝒑𝒐𝒊𝒏𝒕 𝒘𝒉𝒆𝒓𝒆 𝒙 = 𝟏. • Show that the tangent line to y = 3x2 - 12x + 1 at the point (2,-11) is horizontal. • Verify that the tangent line to the line y = 2x + 3 at (1, 5) is the line itself.
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- 18. 1. Determine theEquation of the tangent line at the curve: y = x2 + 3x − 1 at x = 1
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- 20. 2. Determine theEquation of the tangent line at the curve: y = x2 − 2 at x = -2
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- 22. 3. Determine theEquation of the tangent line at the curve: y = 2x2 – 4x + 5 at x = 0
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- 24. 4. Determine theEquation of the tangent line at the curve: y = 3x2 – 12x + 1 at x = 1
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- 26. 5. Determine theEquation of the tangent line at the curve: f(x) = 2x2 – 3x at x = 0
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- 28. 6. Determine theEquation of the tangent line at the curve: 𝒚 = 𝒙 + 𝟗 𝒂𝒕 𝒙 = 𝟎.
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- 30. 7. Determine theEquation of the tangent line at the curve: 𝒚 = 𝟐𝟓 − 𝒙𝟐𝒂𝒕 𝒙 = 𝟑
- 31. Answer y = − 𝟑 𝟒 𝒙+ 𝟐𝟓 𝟒
- 32. 8. Determine theEquation of the tangent line at the curve: 𝒚 = 𝒙𝟐 + 𝒙 𝒂𝒕 𝒙 = 𝟏
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- 34. 9. Determine theEquation of the tangent line at the curve: 𝒚 = 𝟐𝒙𝟐 − 𝟒𝒙 + 𝟓 𝒂𝒕 𝒙 = 𝟏
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- 36. 10. Find thepoint of tangency of the given tangent line to the graph of the function. 𝒇 𝒙 = 𝒙𝟐 − 𝒌𝒙, 𝒚 = 𝟒𝒙 − 𝟗
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