Calculating Limits, Derivatives by Definition, Trigo, Chain Rule
TUTORIAL
CALCULATING LIMITS | Find | | | 1. | limx→1x2-1x-1 | 2. | limt→0t2+9-3t2 | 3. | limx→1x2+x-2x2-3x+2 | 4. | limx→11x-1-2x2-1 | 5. | limx→2x-2x-2 | 6. | limx→∞3x2-x-25x2+4x+1 | 7. | limx→∞x2+1-x | 8. | limx→∞x2+x3-x | 9. | limx→∞2x-13-2x32-3x3 | 10. | limx→∞x2+3x-2+x5x |
Answers : 2, 16, -3,12 , -1 ,35, 0 , -∞, -2 ,25
DERIVATIVES (DIFF. FORMULAS, TRIGO, FUNCTIONS & CHAIN RULE) By using the First Principle Method, find the derivatives of the following. | 1. | y=5x | 2. | fx=5x-1 | | Differentiate the function | | | 1. | Vx=2x3+3x4-2x | 2. | Fy=1y2-3y4y+5y3 | 3. | y=x31-x2 | 4. | y=t23t2-2t+1 | 5. | Suppose that f5=1, f'5=6, g5=-3 and g'5=2Find the following values a) fg'5 b) fg'5 c) gf'5 | 6. | Fx=x4+3x2-25 | 7. | y=cos(a3+x3) | 8. | fx=2x-34x2+x+15 | 9. | y=sin2x+1 | 10. | y=tan4x | 11. | y=sinxcosx | 12. | y=sintan2x | 13. | y=x+x | 14. | y=sinsinsinx | 15. | y=a3+cos3x , where a is constant | 16. | y=xsin1x | 17. | y=x2+1x2-13 | 18. | y=x2+1-3x53 | 19. | gx=2rsinrx+np , where p, n, r are constants | 20. | gt=tan5-sin2t | 21. | ddx5x3-x47 | 22. | y=x28+x-1x4 | 23. | y=sectanx | 24. | fθ=sinθ1+cosθ2 | 25. | r=sinθ2cos2θ | 26. | y=sin2πt-2 | 27. | y=4x+34x+1-3 | 28. | Find the slope of the line tangent to the curve y=sin5x at the point where x=π3 |
Answer :
1. V'x=14x6-4x3-6, 2. F'y=5+9y4+14y2 , 3. y'=x23-x21-x22 , 4. y'=2t(1-t)3t2-2t+12 ,
5. a) -16 b) -209, c) 20 6. 10xx4+3x2-242x2+3 7. -3x2sina3+x3
8. 2x-33x2+x+1428x2-12x-7 11. cosxcosxcosx-xsinx
12. 2sec22xcostan2x 13. 2x+14x x+x 14. cosxcossinsinxcossinx 15. -3sinxcos2x
16. –cos1xx+sin1x 17. -12xx2+12x2-14 18. 3x2+1-3x522x-151-3x4
19. 2pr2cosrx2rsinrx+np-1 20. -2cos2tsec25-sin2t 21. 75x3-x4615x2-4x3
22. 4x28+x-1x3x4+1+1x2 23. sectanxtantanxsec2x 24. 2sinθ1+cosθ2
26. 2πsinπt-2cosπt-2 25. 2θcos2θcosθ2-2sin2θsinθ2 27. 4x+33x+144x+7 28.
CALCULATING LIMITS | Find | | | 1. | limx→1x2-1x-1 | 2. | limt→0t2+9-3t2 | 3. | limx→1x2+x-2x2-3x+2 | 4. | limx→11x-1-2x2-1 | 5. | limx→2x-2x-2 | 6. | limx→∞3x2-x-25x2+4x+1 | 7. | limx→∞x2+1-x | 8. | limx→∞x2+x3-x | 9. | limx→∞2x-13-2x32-3x3 | 10. | limx→∞x2+3x-2+x5x |
Answers : 2, 16, -3,12 , -1 ,35, 0 , -∞, -2 ,25
DERIVATIVES (DIFF. FORMULAS, TRIGO, FUNCTIONS & CHAIN RULE) By using the First Principle Method, find the derivatives of the following. | 1. | y=5x | 2. | fx=5x-1 | | Differentiate the function | | | 1. | Vx=2x3+3x4-2x | 2. | Fy=1y2-3y4y+5y3 | 3. | y=x31-x2 | 4. | y=t23t2-2t+1 | 5. | Suppose that f5=1, f'5=6, g5=-3 and g'5=2Find the following values a) fg'5 b) fg'5 c) gf'5 | 6. | Fx=x4+3x2-25 | 7. | y=cos(a3+x3) | 8. | fx=2x-34x2+x+15 | 9. | y=sin2x+1 | 10. | y=tan4x | 11. | y=sinxcosx | 12. | y=sintan2x | 13. | y=x+x | 14. | y=sinsinsinx | 15. | y=a3+cos3x , where a is constant | 16. | y=xsin1x | 17. | y=x2+1x2-13 | 18. | y=x2+1-3x53 | 19. | gx=2rsinrx+np , where p, n, r are constants | 20. | gt=tan5-sin2t | 21. | ddx5x3-x47 | 22. | y=x28+x-1x4 | 23. | y=sectanx | 24. | fθ=sinθ1+cosθ2 | 25. | r=sinθ2cos2θ | 26. | y=sin2πt-2 | 27. | y=4x+34x+1-3 | 28. | Find the slope of the line tangent to the curve y=sin5x at the point where x=π3 |
Answer :
1. V'x=14x6-4x3-6, 2. F'y=5+9y4+14y2 , 3. y'=x23-x21-x22 , 4. y'=2t(1-t)3t2-2t+12 ,
5. a) -16 b) -209, c) 20 6. 10xx4+3x2-242x2+3 7. -3x2sina3+x3
8. 2x-33x2+x+1428x2-12x-7 11. cosxcosxcosx-xsinx
12. 2sec22xcostan2x 13. 2x+14x x+x 14. cosxcossinsinxcossinx 15. -3sinxcos2x
16. –cos1xx+sin1x 17. -12xx2+12x2-14 18. 3x2+1-3x522x-151-3x4
19. 2pr2cosrx2rsinrx+np-1 20. -2cos2tsec25-sin2t 21. 75x3-x4615x2-4x3
22. 4x28+x-1x3x4+1+1x2 23. sectanxtantanxsec2x 24. 2sinθ1+cosθ2
26. 2πsinπt-2cosπt-2 25. 2θcos2θcosθ2-2sin2θsinθ2 27. 4x+33x+144x+7 28.