PHY 2048C Mathematical Review

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.

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

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

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

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 - 5.0 = 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

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 + 8.0 = 0

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 + 8.0 = 0 Solution: x = -0.91, y = -0.36

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

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:

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

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

2. Trigonometry Example: A fisherman determines that the depth of a lake is 2.25 m when he is 14.0 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:

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

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)

2. Trigonometry Pythagorean theorem:

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

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.

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

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

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

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

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

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

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:

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

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

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

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

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:

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

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

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

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

3. Integral Calculus Common Integrals.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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:

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.):

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!