Course Information

Course

Subject:	Mathematics
Topic:	Political Education
Views:	63,473

Educator

Name:	Math Doctor Bob (Robert Donley)
Type:	Individual

Visit Official Website

Please support this content provider by Donating Now

Donate to Robert Donley

Precalculus Concepts and Exam Problems

Video Lectures

Displaying all 43 video lectures.

Lecture 1 Play Video	Example of Arithmetic Sequence Precalculus: An arithmetic sequence has 20th term equal to 101 and common difference equal to 3. Find the 45th term of the sequence.
Lecture 2 Play Video	Example of Arithmetic Sum Precalculus: Find the sum 7 + 11 + 15 + ... + 227. We identify the terms as belonging to an arithmetic sequence, and show how to derive a formula for the sum.
Lecture 3 Play Video	Geometric Proof of Integer Sum Formula Precalculus: We give an alternative proof that 1+ 2 + ... + n = n(n+1)/2. Our motivation is to explain the 1/2 factor in terms of the area of a triangle.
Lecture 4 Play Video	Example of Geometric Sum Precalculus: Compute the sum 2+ 6 + 18 + ... + 1458. These terms belong to a geometric sequence. A geometric sum formula applies.
Lecture 5 Play Video	Example of Proof By Induction Precalculus: Prove the following statement by induction: 5 + 8 + 11 + ... + (3n+2) = n(3n+7)/2. We review the method using this special case.
Lecture 6 Play Video	Example of Proof by Induction 2: 3 divides 5^n - 2^n Precalculus: We consider a proof by induction with divisibility. Show for all n gt 1 that 3 divides 5^n-2^n. The bookkeeping i simplified by using the definition of divisible. We also show where this class of equations come from.
Lecture 7 Play Video	Example of Proof by Induction 3: n! less than n^n Precalculus: Using proof by induction, show that n! is less than n^n for n greater than 1. We use the binomial theorem in the proof. Also included is a direct proof.
Lecture 8 Play Video	Example of Binomial Theorem Precalculus: Find the first three terms of (x-2y)^8. We use three methods: the binomial theorem, counting, and Pascal's Triangle.
Lecture 9 Play Video	Example of a Piecewise-Defined Function Precalculus: Sketch the graph of the piecewise-defined function f(x) = { x for x leq -3, 4 - x^2 for -3 less than x less than 2, -x for x geq 2}. Then sketch the graph of \|f(x)\| and write \|f(x)\| as a piecewise-defined function.
Lecture 10 Play Video	Domain of sqrt(x+1) - 1/sqrt(9-x^2) Precalculus: Find the domain of the function f(x) = sqrt(x+1) - 1/sqrt(9-x^2). We obtain the domain as the intersection of the domains of two functions.
Lecture 11 Play Video	Example of Composition of Functions Precalculus: Let H(x) = (x-1)^2 - sqrt(x-1). Express H(x) as a composition of two functions (f circ g)(x). Find the domain of H(x) and the composition (g circ f)(x).
Lecture 12 Play Video	Graph of y = -2sqrt(x+4) - 3 Precalculus: Sketch the graph of y = -2sqrt(x+4) - 3 using the graph of y = sqrt(x). We show the graph after each transformation, and track three points through each stage.
Lecture 13 Play Video	Example of Projectile Motion Precalculus: A toy rocket is launched from a 10 ft high platform with an initial velocity of 96 ft/sec. If the height is given by the function h(t) = - 16t^2 +96t +10 ft, find the maximum height attained. We complete the square to find the vertex of graph of h(t), a parabola.
Lecture 14 Play Video	Inverse Function for f(x) = (x-2)/(x+2) Precalculus: Let f(x) = (x-2)/(x+2). Show that f is one-one and sketch the graph of f^{-1}. The find a formula for f^{-1} and verify. We note how to switch four points and the asymptotes from the graph of f to the graph of f^{-1}.
Lecture 15 Play Video	Using Synthetic Division to Evaluate a Polynomial Precalculus: Use synthetic division to evaluate P(7), where P(x) = 3x^5 -22x^4 + 8x^3 - 40x + 11. P(7) is the remainder from division of P(x) by (x-7).
Lecture 16 Play Video	Fast Factorization of Polynomial Precalculus: Let p(x) = x^4 - 6x^3 +13x^2 + ax + b. We are given that x^2-3x+2 divides p(x). Find a and b, and show that P(x) ge 0 for all x. We note three approaches and show the work for two.
Lecture 17 Play Video	Multiplying Complex Numbers Precalculus: We describe multiplication of complex numbers. Examples given are (a) 2i (3 - 4i), (b) (5 + 2i) ( 3 + 4i), (c) (x + 1 + i) (x - 3i), and (d) (1/2 + sqrt(3)/2 i) ( sqrt(3)/2 + 1/2 i).
Lecture 18 Play Video	Complex Conjugate Roots of a Real Polynomial Precalculus: Find a polynomial with real coefficients of degree 4 that has complex roots 1+i and 3i. We use the fact that the roots of a real polynomial occur in complex conjugate pairs if there is a nonzero imaginary part.
Lecture 19 Play Video	Factoring 3x^4-2x^3+3x-2 Over The Complex Numbers Precalculus: Factor the polynomial P(x) = 3x^4-2x^3+3x-2 completely over the complex numbers. Techniques include grouping and synthetic division.
Lecture 20 Play Video	Sketch of Rational Function Precalculus: Sketch the graph of the rational function r(x) = (x^2 - 3x + 4)/(x^3 - 6x^2 - 4x + 24). Label the horizontal and vertical asymptotes, and the x- and y-intercepts. Find all regions where the function is positive or negative. Determine the end behavior of the function.
Lecture 21 Play Video	Partial Fraction Expansion of (x^4+x^3+1)/(x(x^2+1)^2) Precalculus: Find the partial fraction expansion of the rational function (x^4+x^3+1)/(x(x^2+1)^2). This expansion requires two terms for the irreducible quadratic x^2+1.
Lecture 22 Play Video	Evaluation of 3ln(2e^2) + 2ln(3/e) - ln(72e) Precalculus: Evaluate the expression with natural logarithms: 3ln(2e^2) + 2ln(3/e) - ln(72e). We show two methods: one where we expand all terms, and one where we put all terms under a single logarithm.
Lecture 23 Play Video	Domain of ln((x+1)/(x-1)) +ln(x-1)-ln(x+1) Precalculus: Find the domain of the function f(x) = ln((x+1)/(x-1))+ln(x-1)-ln(x+1). First we show why simplifying the function leads to problems.
Lecture 24 Play Video	Solving an Exponential Equation Precalculus: Find all x that solve the equation e^{2x} + e^x - 12 = 0. We substitute y = e^x, factor, and use natural logarithm to isolate x.
Lecture 25 Play Video	Memorizing Trig Values: Three Easy Pieces Trigonometry: We present a scheme for memorizing cosine and sine for the important angles (multiples of pi/4, pi/3, and pi/6). The three main parts are the 30-60-90 triangle, the CAST method, and the rectangular grouping of the special angles.
Lecture 26 Play Video	Trig Values for Multiples of pi/5 Trigonometry: We calculate cosine and sine for the multiples of pi/5. The resulting formulas and relations are connected to the Golden Ratio.
Lecture 27 Play Video	Examples of Values of Trig Functions Precalculus: Find the values of all six trig functions for the angle theta where theta is in Quadrant 4 and sin(theta) = -7/25. We give two approaches to the solution.
Lecture 28 Play Video	Calculating Trig Values Precalculus: Calculate the following trig values: (a) sin(5pi/4), (b) sec(pi), and (c) cot(-30 degrees). We use reference angles and the CAST method.
Lecture 29 Play Video	Examples of Inverse Trig Evaluations Precalculus: Evaluate the following inverse trig expressions: (a) tan^{-1}(-sqrt(3)), (b) sin^{-1}(sin(5pi/4)), and (c) sec(tan^{-1}(3/8)).
Lecture 30 Play Video	Graph of y = -4sin(3x-pi) Precalculus: Sketch the graph of one period of the function y = -4sin(3x-pi). Find the amplitude, the period, and the phase shift.
Lecture 31 Play Video	Graph of y = 2 sec(pi x - pi/3) Precalculus: Sketch one period of the graph of the function y = 2 sec(pi x - pi/3). We review the graph of cos(x) to obtain our sketch.
Lecture 32 Play Video	Example of Converting to Polar Coordinates Precalculus: Convert the following points from rectangular to polar coordinates: (a) (0,-3), (b) (-4,4), and (c) (3, -sqrt(3)). We use the equations r = sqrt(x^2+y^2), x = rcos(theta), and y = rsin(theta).
Lecture 33 Play Video	Example of Graph of Polar Equation Precalculus: Sketch the graph of the polar equation r = 2 - 2sin(theta) in the plane. The resulting graph is a cardioid.
Lecture 34 Play Video	Example of Proof of Trig Identity Precalculus: Prove the trig identity cos(x + y) cos(x-y) = cos^2(x) - sin^2(y). We use the sum angle formula for cosine, the even/oddness of cosine and sine, and the identity cos^2(A) + sin^2(A) = 1.
Lecture 35 Play Video	Example of Double Angle Formulas Precalculus: Compute the sine and cosine of theta = 2 tan^{-1}(-3/4). We first compute the sine and cosine for tan^{-1}(-3/4). Then we apply the double angle formulas for sine and cosine.
Lecture 36 Play Video	Example of Half Angle Formula Precalculus: Compute sine and cosine of 9pi/8. We emphasize the use of the double angle formula for cosine over a half angle formula. In either case, one is required to use CAST to find the sign on each quantity.
Lecture 37 Play Video	Example of Trig Equation Precalculus: Find all solutions to the trig equation [sin(4x)]^3 - 4sin(4x) = 0 for x in [0, 2pi). First we solve y^3 - 4y = 0. Those solutions yield the equations sin(4x) = 0, +2, -2, which we then solve.
Lecture 38 Play Video	Trig Values for Multiples of pi/5 Trigonometry: We calculate cosine and sine for the multiples of pi/5. The resulting formulas and relations are connected to the Golden Ratio.
Lecture 39 Play Video	Example of De Moivre's Theorem Precalculus: Evaluate (1+i)^10. Put final answer in the form x + iy. We use De Moivre's Theorem to compute. This requires a change to polar coordinates. We check the answer by a straightforward computation, noting that (1+i)^2 = 2i.
Lecture 40 Play Video	Cube Roots of -8 Precalculus: Find the cube roots of -8 in the complex numbers, and sketch as points in the complex plane. We employ two approaches: one using deMoivre's Theorem, and the other by factoring z^3 + 8.
Lecture 41 Play Video	Complex Solutions of z^4-4z^2+16=0 Complex Analysis: Find all solution to the equation z^4 - 4z^2 +16 = 0 over the complex numbers. The technique involves the substitution y = z^2 and converting complex numbers into polar form.
Lecture 42 Play Video	Example of Law of Sines Precalculus: A triangle is given with two angles, 30 degrees and 15 degrees, and one side of length 10 (see picture). Find the lengths of the missing sides using the law of sines.
Lecture 43 Play Video	Example of Law of Cosines Precalculus: A triangle has one angle with 60 degrees and opposite side of length 10. The other sides have lengths x and 3. Find x using the Law of Cosines. We also find an alternative solution using right triangles. Then the Law is derived and examined in various cases.