# Precalculus Definition and 43 Discussions

In mathematics education, precalculus or college algebra is a course, or a set of courses, that includes algebra and trigonometry at a level which is designed to prepare students for the study of calculus. Schools often distinguish between algebra and trigonometry as two separate parts of the coursework.

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1. ### Inequalities: x^2 > 900

We can also write a general inequality ##x^2>a## where a is a number. If ##x^2= a##, then ##x= \pm \sqrt a## which means ##x= \sqrt a## or ##-\sqrt a##. But in this case i don’t think it will be ##x > \pm \sqrt a## because if we take ##0## , it’s greater than a negative but in the original...
2. ### Find all possible solutions of x^3 + 2 = 0

I actually know one way to solve, ##x ^ 3 + 2 = 0## ##x ^ 3 + \left (\sqrt[3] 2\right) ^ 3 = 0## ##\left(x + \sqrt[3] 2\right) \left(x ^ 2 - x\sqrt[3] 2 + \left(\sqrt[3] 2 \right)^2\right) =0## ##x + \sqrt[3] 2 = 0, x = -\sqrt[3] 2## ##x ^ 2 - x\sqrt[3] 2 + \left(\sqrt[3] 2\right)^2 = 0, x =...
3. ### Find the values of a real number ##a## for an inequality to hold

Problem statement : Let me copy and paste the problem as it appears in the text. Attempt : From the "Relevant Equations" given above, we can compare to see that ##a-1 = -1## and ##a^2+2=3##. These lead (after some algebra) to the three values of ##\boxed{a=0, \pm 1}##. Issue : The book has...
4. ### The given function is an even or an odd function?

I think the answer is an even function as the function ##x^2## is an even function and thus, is symmetrical w.r.t. Y axis. The question I have is how to do this problem algebraically. I tried to graph some functions on GeoGebra to verify my answer. a) ##y = ln(x^2)## b) ##y = sin(x^2)##...
5. ### Finding the convergence of a parametric series

It is clear that the terms of the sequence tend to zero when n tends to infinity (for some α) but I cannot find a method that allows me to understand for which of them the sum converges. Neither the root criterion nor that of the relationship seem to work. I tried to replace ##\sqrt[n]{n}## with...
6. ### Calculating the half maximum point of a function

This is the form of the function above: I started by equating (1) to 1/2: $$T(\varphi)=\frac{r^{2}+\tau^{2}-2\tau\cos\varphi}{1+\tau^{2}r^{2}-2\tau r\cos\varphi} = \frac{1}{2},$$ which can be rearranged to: $$2r^{2}+2\tau^{2}-1-\tau^{2}r^{2}=2\tau\left[2-r\right]\cos\varphi$$ using...
7. ### Find the derivative of a balloon's circumference

A balloon's volume is increasing at a rate of dV/dt. Express the rate of change of the circumference with respect to time (dc/dt) in terms of the volume and radius. Homework Equations Vsphere = (4/3)(π)(r^3) C = (2)(π)(r) The Attempt at a Solution [/B] My strategy was to come up with two...
8. ### Hi, I have a quick question about graph transformations.

One of my homework questions said "Explain how to obtain f(x)=-(3+x)^2+1 from the graph of y=x^2." I know somehow you need to move the graph right 3, reflect about the x-axis, and move up one, but I don't know how to factor and manipulate the equation to show this.
9. ### Find the center of a circle given a tangent line & point

Homework Statement "Find the center and radius of the circle that passes through A(1,1) and is tangent to the line y=2x-3 at the point B(3,3). (Picture of the graph: https://imgur.com/a/0wAnqcU) Homework Equations Here's a link: https://imgur.com/a/y71Z9GY The Attempt at a Solution Soo, I've...
10. ### Hi, I have a quick question about inverse functions.

One of our homework problem asks: If f is a one-to-one function such that f(-3)=5 , find x given that f^-1 (5)=3x-1. Here's how I attempted to solve the problem: -3=3x-1 3x=-2 x=-2/3 Is this the correct way to solve the problem?
11. ### Matrix help — Solve this matrix equation

Homework Statement Homework Equations The Attempt at a Solution I know I would have to do something with my calculator and I tried to solve like solving an equation for C, but not sure. I put all the matrices in my calculator. I then subtracted the first matrix to the other side then...
12. ### Vectors: Proving the value of a•(b x c)

Homework Statement Given that vector a = (1, 2, -5), b = (-12, 41, 75) and c = a + 2b, explain why (without doing any calculations whatsoever) the value of a•(b x c) = 0 Homework Equations No specific equations, as the question asks for the value without making any calculations. This problem...
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### B Why do equations with two distinct variables with 2 distinct

My question is about basic algebra. I am thinking about the "why" here and I'm looking for an intuitive answer. If you have the following equations: =>S+U=90 =>S+U=90 and =>40S+25U=2625 =>40S+25U=2625 you can then rewrite S=90−U and then substitute. Now you have a single equation with one...
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### Transforming Piecewise Functions

Homework Statement The piece wise function is -x-2, x<-1 x^2-3x, -1≤ x ≤5 3x+5, x>5 The problem is to transform the function with...
15. ### Calculus Need help preparing for Courant's Calculus

Hello, Just to give you some background info, I'm a software developer looking to make a career switch to Mechanical Engineering (At least I think so). My interests lie in design, complex systems, maglev, rockets and new forms of space propulsion. Before I really commit myself to University, I...
16. ### Find ground speed and direction of airplane

Homework Statement [/B] "An airplane is flying on a bearing of 340 degrees at 400mph. A wind is blowing at a bearing of 320 degrees at 30mph. Find ground speed and direction of the plane." Homework Equations vx=vcosϕ vy=vsinϕ The Attempt at a Solution [/B] First, my teacher told us to change...
17. ### Logarithm question using compound interest formula

Homework Statement A colony of ants will grow by 12% per month. If the colony originally contains 2000 ants how long will it take for the colony to double in size? Answer - 6.12 months Homework Equations A = P(1+r/n)nt The Attempt at a Solution r = 12% = 0.12 n = 12 P = 2000 A = 4000 t = ...
18. ### Solving Rational Functions

Homework Statement Sketch the graphs of the following functions and show all asymptotes with a dotted line y = (2x - 6)/ (x2-5x+4) i) Equation of any vertical asymptote(s) ii) State any restrictions or non-permissible value(s) iii) Determine coordinates of any intercept(s) iv) Describe the...
19. ### Find the limit of the expression

Homework Statement $$\lim_{x\to\infty} \left(\frac{n^2+2n+1}{n^2+2n+3}\right)^{\frac{2n^2}{n+1}}$$ Homework Equations 3. The Attempt at a Solution [/B] I tried ##\lim_{x\to\infty} \left(\frac{n^2+2n+3-2}{n^2+2n+3}\right)^{\frac{2n^2}{n+1}}=## ##\lim_{x\to\infty}...
20. ### Foundations I have complied a list of textbooks about Trig, Algebra, Etc

I am planning to self-teach myself Trigonometry and all the other required fields before jumping into calculus. I have compiled a list of books that I have researched and would like your opinions and recommendations. Books: Precalculus Mathematics in a Nutshell: Geometry, Algebra, Trigonometry...
21. ### Can't find all the zeroes of a polynomial

Homework Statement Help i have a homework quiz done and i simply can't find out how to do the 3rd problem as we haven't even learned how to do it or maybe my notes aren't good or something , however I am close to an A in the class and this would help bring it closer. It asks me: "Find all the...
22. ### I can't understand this problem

Homework Statement Ok so this isn't really a problem, more like a problem set, I'm not sure if I'm able to understand it yet. The context is determining all the primitive pythogrean triples Letting x = a/c and y = b/c, we see that (x, y) is a point on the unit circle with rational number...
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### What do I need to master before I start calculus?

I want to start learning calculus I. But, I don't know what courses I need to finish before starting precalculus. any help would be appreciated.
24. ### Homework help on vectors please

Homework Statement http://imgur.com/a/Yq8pW Homework Equations projection u onto v: ((u x v)/(||v||^2)) x v Unit vector: u/||u|| The Attempt at a Solution For number 2, I absolutely do not know how to set up the problem. I do not know what vectors to use, I assumed F vector to be <0.00375...
25. ### A problem from polynomials

Homework Statement [/B] Th value of 'a' for which the equation x3+ax+1=0 and x4+ax+1=0 have a common root is? Homework Equations The Attempt at a Solution i initially thought of subtracting both the equations and then finding x and substituting back in the equation but it did not work.
26. ### A permutation and combination problem

.The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices (0, 0), (0, 41) and (41, 0), is (1) 901 (2) 861 (3) 820 (4) 780 my attempt: for this to be true i know that sum of x and y coordinate should be 41 but i don't know how to proceed.
27. ### Trig Identities

I have re-post this forum as I should have paid closer attention to rules. I apologized for that. Homework Statement 1) The expression tan^3 θ + sinθ/cosθ is equal to: (a) cot θ (b) tan θ sec^2 θ (c) tan θ (d) sin θ tan θ (e) tan θ csc^2 θ 2) Simplify (cos θ/1+ sin θ - cosθ/sinθ-1)^-1...
28. ### Find the sets of real solutions

[b[1. Homework Statement [/b] ##|4^{3x}-2^{4x+2}*3^{x+1}+20*12^x*3^x| \ge 8*6^x(8^{x-1}+6^x)## The sets containing the real solutions for some numbers ##a, b, c, d,## such that ##-\infty < a < b < c < d < +\infty## is of the form ##(-\infty, a] \cup [b, c] \cup [d, +\infty)##. Prove it by...
29. ### Find real solutions to an equation

Homework Statement log(cosx)sinx = 4*log(sinx)cosx Homework Equations 3. The Attempt at a Solution i tried to solve it and uploaded my work but my last part reads (cosx)^2 = sinx (cosx)^(-2) = sinx its weird that i don't see where i have made a mistake[/B]
30. ### Area of a triangle under a curve

Homework Statement The tangent line of a curve y=e^(-x) intercepts the axises at points A and B. What is the maximum area of a triangle AOB considering O as the origin. Homework Equations Ar= xy/2 The Attempt at a Solution Derivative of this function is y'=-e^(-x) I took the formula of the...
31. ### Trigonometric problem

Homework Statement cos(x - 3p/2) = - 4/5 p <x< p/2 sin(x/2)*cos(5p/4)= ? Homework Equations The Attempt at a Solution I made it as far as to determine that sinx= 4/5 and cosx = - 3/5 but can't seems to progress any further. I am looking for an easier way to find the solution without having to...
32. ### How to solve this logarithmic equation?

Homework Statement For clarification a have posted the equation below as a picture file.[/B] Homework Equations log(a*b)= loga + logb log(a/b) = loga-logb log(a^n) = nloga[/B] The Attempt at a Solution I don't know how to start. I can't remember the rule for powering the logarithms if there...
33. ### Finding the quadratic equation that suits the solutions

Homework Statement if x1 and x2 are solutions to 1) x² + x + 1 = 0, then 2) y1 = ax1 + x2 and 3) y2 = x1 + ax2 are solutions to which quadratic equation? Homework Equations ax² + bx + c = 0 x1∕2 = (−b ± √(b² - 4ac))/2a The Attempt at a Solution Well, firstly i solved for x1 and x2 getting...
34. ### Calculus Best math textbook covering these topics?

Hi everyone! I am starting to prepare for my admission test into college next year and I am looking for a best textbook which covers these topics ( with an adition of other it may also include but not disclude these) Polynomials Functions ( linear, quadratic ) Algebraic equations and...
35. ### Combinatorics question: Identical/nonidentical

Homework Statement and attempt at a solution[/B] If 5 gifts are to be given among 8 children: a) if the gifts are identical (indistinguishable) and no child can receive more than 1 gift, there are 8P5 ways b) if the gifts are non-identical (distinguishable) there are 5!(8P5) ways In a), the...
36. ### R^2=9cos(3Ө) Convert to rectangular cordinates

Homework Statement Homework Equations The Attempt at a Solution -In order for me to figure out this problem I had to reverse the equality to isolate the Ө on the left side making the new equation 9cos(3Ө)=r^2. The first thing I’m going to do is change cos(3Ө) in terms of cos and sin. That...
37. ### Foundations Theoretical Books on Mathematics

What are some rigorous theoretical books on mathematics for each branch of it? I have devised a fantastic list of my own and would like to hear your sentiments too. Elementary Algebra: Gelfand's Algebra Gelfand's Functions & Graphs Burnside's Theory of Equations Euler's Analysis of the...
38. ### How do I teach myself precalculus up to calculus?

What are some good textbooks, websites, question papers and other sources of information that will help me teach myself precalculus up to calculus? How long will this take until I have a good grasp of the knowledge?
39. ### Rational and irrational numbers

Homework Statement Determine a positive rational number whose square differs from 7 by less than 0.000001 (10^(-6)) Homework Equations - The Attempt at a Solution Let p/q be the required rational number. So, 7> (p/q)^(2) > 7-(0.000001) ⇒ √(7) > p/q > √(7-.000001) ⇒√(7) q> p >...