Graph oh functions
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The document contains graphs of trigonometric functions such as sin(x), tan(x), and cot(x) with their domains and ranges. It also discusses properties of these functions including that they have: - Maxima and minima occurring at specific values of x where the derivative is zero and the tangent line is parallel to the x-axis. - The slope of the tangent line to the derivative is negative at maxima.
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Graph oh functions
- 1. I y = f ( x) = x y I I I I I I I I I I I x’ I -4 -3 -2 -1 0 1 2 3 4 x I I I Y’ I
- 2. I y = f ( x) = IxI y I I I I I I I I I I I I x’ -4 -3 -2 -1 1 2 3 4 0 x I I I Y’ I
- 3. I y Y = f ( x) = 1/x I I x’ I I I I I I I I I I -4 -3 -2 -1 0 1 2 3 4 x I I I Y’ I
- 4. I y Y = f ( x) =[ x ] I I I I I I I I I I I I x’ -4 -3 -2 -1 1 2 3 4 0 x I I I Y’ I
- 5. I y I I I I I I I I I I I x’ I -4 -3 -2 -1 0 1 2 3 4 x Y = f ( x) = x 2 I I I Y’ I
- 6. x 0 / 6 / 3 /2 2 /3 5 /6 sin x 0 5 87 1 87 5 0 1 .5 I I I I I I I I I I I I I I I I I I - π 5π/6 -2π/3 - π/2 -π/3 - π/6 0 π/6 π/3 π/2 2π/3 5π/6 π .5 1
- 7. x 0 /6 /3 /2 2 /3 5 /6 sin x0 5 87 1 87 5 0 1 .5 I I I I I I I I I I I I I I I I I I - π 5π/6 -2π/3 - π/2 -π/3 - π/6 0 π/6 π/3 π/2 2π/3 5π/6 π .5 1
- 8. x 0 / 6 / 3 /2 2 /3 5 /6 7 / 6 4 / 3 3 / 2 5 / 3 11 / 6 2 sin x 0 5 87 1 87 5 0 5 87 1 87 5 0 1 I I .5 I I I I I I I I I I I I I I I I I I I 0 π/6 π/3 π/2 2π/3 5π/6 π 7π/6 4π/3 3π/2 5π/3 11π/6 2π .5 I 1 I
- 9. x 0 /6 /3 /2 2 /3 5 /6 7 / 6 4 / 3 3 / 2 5 / 3 11 / 6 2 sin x 0 5 87 1 87 5 0 5 87 1 87 5 0 1 I I .5 I I I I I I I I I I I I I I I I I I I - 2π -11π/6 -5π/3 -3π/2 -4π/3 -7π/6 -π - 5π/6 -2π/3 - π/2 - π/3 -π/6 0 .5 I 1 I
- 10. x 0 /6 /3 /2 2 /3 5 /6 sin x 1 87 5 0 5 87 1 1 .5 I I I I I I I I I I I I I I I I I I - π 5π/6 -2π/3 - π/2 -π/3 - π/6 0 π/6 π/3 π/2 2π/3 5π/6 π .5 1
- 11. x 0 /6 /3 /2 2 /3 5 /6 sin x 1 87 5 0 5 87 1 1 .5 I I I I I I I I I I I I I I I I I I - π 5π/6 -2π/3 - π/2 -π/3 - π/6 0 π/6 π/3 π/2 2π/3 5π/6 π .5 1
- 12. 1 I I I I I I I I I I -2π -3 π/2 -π - π/2 0 π/2 π 3 π/2 2π -1
- 13. x 0 / 6 / 3 /2 0 /2 0 2 /3 5 /6 7 / 6 4 / 3 3 / 2 0 3 / 2 0 5 / 3 11 / 6 2 tan x 0 58 1 73 1 73 58 0 58 1.73 1 73 58 0 1 I I .5 I I I I I I I I I I I I I I I I I I I 0 π/6 π/3 π/2 2π/3 5π/6 π 7π/6 4π/3 3π/2 5π/3 11π/6 2π 5π/2 .5 I 1 I
- 14. x 0 tan x 0 1 I I .5 I I I I I I I I I I -2π -3 π/2 -π - π/2 0 π/2 π 3 π/2 2π .5 I 1 I
- 15. x 0 /6 /3 /2 2 /3 5 / 6 0 0 7 / 6 4 / 3 3 / 2 5 / 3 11 / 6 2 0 cot x 1 73 58 0 58 1 73 1 73 .58 0 58 1 73 1 I I .5 I I I I I I I I I I I I I I I I I I I 0 π/6 π/3 π/2 2π/3 5π/6 π 7π/6 4π/3 3π/2 5π/3 11π/6 2π 5π/2 .5 I 1 I
- 16. 1 I I .5 I I I I I I I I I I -2π -3 π/2 -π - π/2 0 π/2 π 3 π/2 2π .5 I 1 I
- 17. -2π I Goes tp + infinity I -3 π/2 I Goes tp + infinity -π Goes tp - infinity I - π/2 Goes tp - infinity 1 1 0 I Goes tp + infinity I π/2 Goes tp + infinity I π Goes tp - infinity I 3 π/2 Goes tp - infinity I 2π I
- 18. Goes tp + infinity -2π II Goes tp + infinity -3 π/2 Goes tp - infinity I -π Goes tp - infinity I - π/2 Goes tp + infinity 1 1 0 II π/2 Goes tp + infinity Goes tp - infinity I π Goes tp - infinity Goes tp + infinity I 3 π/2 I 2π Goes tp + infinity I
- 19. I 1 f ( x) sin x I I I I I I I
- 20. I y f(x) = x I x y e I I I I I I I I I I I x’ -4 -3 -2 -1 (1,0) 2 3 4 0 x I y log e x I I Y’ I
- 21. I y = f ( x) = x y I I I I I I I I I I I x’ I -4 -3 -2 -1 0 1 2 3 4 x When x increases f( x) also increases. f(Y == x is=strictly increasing function x) f( x) x is continuous function. I I I Y’ I
- 22. ….- 3 π/2 , π/2 , 5 π/2 …. Are maxima as at these values of x ,f(x) is maximum. 1 I I I I I I I I I I -2π -3 π/2 -π - π/2 0 π/2 π 3 π/2 2π 5π/2 -1 -
- 23. ….- π/2 , 3π/2 , …. are minima as at these values of x, f(x) is minimum. 1 I I I I I I I I I I -2π -3 π/2 -π - π/2 0 π/2 π 3 π/2 2π 5π/2 -1
- 24. At both maxima at minima the Tangents are parallel to x – axis. 1 I I I I I I I I I I -2π -3 π/2 -π - π/2 0 π/2 π 3 π/2 2π -1
- 25. At both maxima at minima the Derivative of the function is zero. Or f x 0 1 I I I I I I I I I I -2π -3 π/2 -π - π/2 0 π/2 π 3 π/2 2π f -1
- 26. Slope of the tangent is zero of the tangent is zero Slope 1 I I I I I I I I I I -2π -3 π/2 -π - π/2 0 π/2 π 3 π/2 2π -1 Slope of the tangent is zero
- 27. ' f ( x) cos x At maxima slope of the tangent to the Derivative is negative. 1 I I I I I I I I I I -2π -3 π/2 -π - π/2 0 π/2 π 3 π/2 2π -1 ….- maxima as wellπ/2 minima the slope At 3 π/2 , π/2 , 5 as …. Are maxima ….- π/2 , 3π/2 , …. Are minima of the Tangents are zero.