1. PHY 2048C
Mathematical Review

2. PHY 2048C Mathematical Review Note:
The following mathematical concepts will be used extensively in this physics course. Meeting the prerequisites indicates that you have mastered the following material, however some of you may have not seen this material in a few semesters. If you are unsure of whether or not you have the appropriate mathematical background necessary to succeed in Physics with Calculus I, I encourage you take a look at the following review and to work out the practice examples on your own. Solutions are given on the following slide.

3. Quadratic Equations. A = 4.9, B = -8.0, C = -3.6 1. Algebra
If an equation can be put in the form:
Ax2+Bx+C = 0
Then the two solutions are given by the quadratic formula:
Example:
Solve for t:
A = 4.9, B = -8.0, C = -3.6

4. 1. Algebra
Your Turn: Solve for q.
2.8q = 7.0 – 4.5q2

5. Your Turn: Solve for q. 2.8q = 7.0 – 4.5q2 Solution: q = -1.60, 0.97
1. Algebra
Your Turn: Solve for q.
2.8q = 7.0 – 4.5q2
Solution: q = -1.60, 0.97

6. Solving a system of multiple equations.
1. Algebra
Solving a system of multiple equations.
If the n independent equations containing the less than n unknown variables each represented in each equation, the system can be solved for all n unknowns.
Example:
Solve the system of equations for x and y:
(1) 2.0x - 1.7y = 4.1
(2) 3.1x = 13.0y
Solution: x = 2.16, y = 0.13
Steps:
1. Solve equation (2) for y: y = (3.1x – 5.0)/13.0
2. Plug this solution for y into equation (1) : 2.0x - 1.7(3.1x – 5.0)/13.0 = 4.1
3. Solve for x: x = 2.16
4. Knowing x, solve for y: y = 0.13

7. 1. Algebra
Your Turn: Solve the system of equations for x and y.
(1) 5.0x - 4.0y = 3.1
(2) 6.0x – 7.2y = 0

8. 1. Algebra
Your Turn: Solve the system of equations for x and y.
(1) 5.0x - 4.0y = 3.1
(2) 6.0x – 7.2y = 0
Solution: x = -0.91, y = -0.36

9. 2. Trigonometry
The trigonometric functions sine, cosine, and tangent are defined as the following ratios of sides of a right triangle:

10. 2. Trigonometry Example:
Determine the height of a building if, as you stand 67.2 m from the building you can just see the Sun over the top of the building at an altitude angle of 50º.
Solution:

11. 2. Trigonometry
The inverse trig functions produce the angle from the ratio of the sides of the right triangle.

12. 0 < θ < π/2 (when using tan-1)
2. Trigonometry
The inverse trig functions produce the angle from the ratio of the sides of the right triangle.
Note: 0 < θ < π (when using sin-1 or cos-1)
0 < θ < π/2 (when using tan-1)
Here θ is given in radians (rad). 1 rad = 1º × 2π/360º

13. 2. Trigonometry Example:
A fisherman determines that the depth of a lake is 2.25 m when he is m from the shore. If the bottom of the lake has a constant slope, determine the angle the bottom of the lake makes with the water’s surface.
Solution:

14. 2. Trigonometry
Your Turn: Solve for θ in the right triangle shown below.
6.32 m
2.00 m
6.00 m

15. 2. Trigonometry
Your Turn: Solve for θ in the right triangle shown below.
6.32 m
2.00 m
6.00 m
Solution:
(0.321 rad)

16. 2. Trigonometry
Pythagorean theorem:

17. R 2.00 m 6.00 m 2. Trigonometry Example:
Determine the length of the hypotenuse of the right triangle shown below, R. R
2.00 m
6.00 m

18. A displacement vector has a magnitude of 175 m and points at
2. Trigonometry
Physics Example:
A displacement vector has a magnitude of 175 m and points at
an angle of 50.0 degrees relative to the x axis. Find the x and y
components of this vector.

19. 3. Differential Calculus
The definition of the derivative.
The slope of the tangent line to a curve. Δx Δt

20. Basic Differentiation Rules.
3. Differential Calculus
Basic Differentiation Rules. 1.
Example: 2.
Example:

21. Basic Differentiation Rules.
3. Differential Calculus
Basic Differentiation Rules. 3.
Example: 4.
Example:

22. More Differentiation Rules.
3. Differential Calculus
More Differentiation Rules. 5.
Product Rule
Example:
Derivative of the second function
Derivative of the first function

23. More Differentiation Rules.
3. Differential Calculus
More Differentiation Rules. 6.
Quotient Rule
Sometimes remembered as:

24. More Differentiation Rules.
3. Differential Calculus
More Differentiation Rules. 6.
Quotient Rule (cont.)
Example:
Derivative of the denominator
Derivative of the numerator

25. Note: h(x) is a composite function.
3. Differential Calculus
More Differentiation Rules.
7. The Chain Rule
Note: h(x) is a composite function.
Another Version:

26. The General Power Rule:
3. Differential Calculus
More Differentiation Rules.
The Chain Rule leads to
The General Power Rule:
Example:

27. Your Turn: Differentiate with respect to x.
3. Differential Calculus
Chain Rule Example.
Your Turn: Differentiate with respect to x.

28. Your Turn: Differentiate with respect to x.
3. Differential Calculus
Chain Rule Example.
Your Turn: Differentiate with respect to x.
Solution:

29. Chain Rule Example. 3. Differential Calculus Example:
Substitute for u:

30. Anti-Derivatives, integrals, and the fundamental theorem of calculus.
3. Integral Calculus
Anti-Derivatives, integrals, and the fundamental theorem of calculus.
An anti-derivative of a function f(x) is a new function F(x) such that
Indefinite integral:
Definite integral:

31. Definite Integral as Area Under the Curve.
3. Integral Calculus
Definite Integral as Area Under the Curve.
Exact Area=

32. Definite Integral with Variable Upper Limit.
3. Integral Calculus
Definite Integral with Variable Upper Limit.
, More “proper” form with “dummy” variable:
Improper Integrals.

33. Example: Determine using the figure below.
3. Integral Calculus
Example: Determine using the figure below.

34. Example: Determine using the figure below.
3. Integral Calculus
Example: Determine using the figure below.
Solution:

35. 3. Integral Calculus
Common Integrals.

36. Common Integrals (Cont.)
3. Integral Calculus
Common Integrals (Cont.)

37. Your Turn: Integrate with respect to x.
3. Integral Calculus
Your Turn: Integrate with respect to x.

38. Your Turn: Integrate with respect to x.
3. Integral Calculus
Your Turn: Integrate with respect to x.
Solution:

39. Your Turn: Integrate with respect to x.
3. Integral Calculus
Your Turn: Integrate with respect to x.

40. Your Turn: Integrate with respect to x.
3. Integral Calculus
Your Turn: Integrate with respect to x.
Solution:

41. Your Turn: Integrate with respect to x.
3. Integral Calculus
Your Turn: Integrate with respect to x.

42. Your Turn: Integrate with respect to x.
3. Integral Calculus
Your Turn: Integrate with respect to x.
Solution:

43. Your Turn: Determine the definite integral:
3. Integral Calculus
Your Turn: Determine the definite integral:

44. Your Turn: Determine the definite integral:
3. Integral Calculus
Your Turn: Determine the definite integral:
Solution:

45. Your Turn: Determine the definite integral:
3. Integral Calculus
Your Turn: Determine the definite integral:

46. Your Turn: Determine the definite integral:
3. Integral Calculus
Your Turn: Determine the definite integral:
Solution:

47. Physics Example: Displacement, Velocity, and Acceleration.
3. Integral Calculus
Physics Example: Displacement, Velocity, and Acceleration.

48. Physics Example: Displacement, Velocity, and Acceleration.
3. Integral Calculus
Physics Example: Displacement, Velocity, and Acceleration.

49. Physics Example: Displacement, Velocity, and Acceleration.
3. Integral Calculus
Physics Example: Displacement, Velocity, and Acceleration.

50. Example: An object experiences acceleration as given by:
3. Integral Calculus
Example: An object experiences acceleration as given by:
where
Determine the velocity and displacement.

51. Example: An object experiences acceleration as given by:
3. Integral Calculus
Example: An object experiences acceleration as given by:
where
Determine the velocity and displacement.
Solution:

52. Example: An object experiences acceleration as given by:
3. Integral Calculus
Example: An object experiences acceleration as given by:
where
Determine the velocity and displacement.
Solution (cont.):

53. You’ve completed the PHY 2048C mathematical review.
Thanks!
Note: On the first day of class, you will be given a quiz based on these concepts. Good luck!