Articles of algebra precalculus

Finding the equation of the normal line

I have a question to find the equations of the tangent line and the normal line to the curve at the given point. I can find the equation for the tangent line easily but I am not sure what a normal line is and there is no example that I can find. $y=x^4 + 2e^x$ […]

If $\sin x + \sin^2 x =1$ then find the value of $\cos^8 x + 2\cos^6 x + \cos^4 x$

If $\sin x + \sin^2 x =1$ then find the value of $\cos^8 x + 2\cos^6 x + \cos^4 x$ My Attempt: $$\sin x + \sin^2 x=1$$ $$\sin x = 1-\sin^2 x$$ $$\sin x = \cos^2 x$$ Now, $$\cos^8 x + 2\cos^6 x + \cos^4 x$$ $$=\sin^4 x + 2\sin^3 x +\sin^2 x$$ $$=\sin^4 x […]

Express each of the following expressions in the form $2^m3^na^rb^s$, where $m$, $n$,$ r$ and $ s$ are positive integers.

I just recently started relearning math as an adult, this should be easy but I have trouble understanding what the actual question is. I am not just looking for the answer to this, I merely wish to understand what the question is asking. Express each of the following expressions in the form $2^m3^na^rb^s$, where $m$, […]

Find the values of $m$ in the 2nd degree equation $mx^2-2(m-1)x-m-1=0$ so that it has one root between $-1$ and $2$

Find the values of $m$ in the 2nd degree equation $mx^2-2(m-1)x-m-1=0$ so that it has only one root between $-1$ and $2$. Like in this almost identical question there are two ways to solve this, one is acknowledging that $f(-1)f(2) \lt 0$, the other applying the theorems for the two possible scenarios $x_1 \lt -1 […]

Proving that for reals $a,b,c$, $(a + b + c)^2 \leq 3(a^2+b^2+c^2)$

Proving that for reals $a,b,c$, $(a+b+c)^2\leq 3(a^2+b^2+c^2)$. This is a homework question and I have no clue where to even start on this. I don’t know if I am just tired or what but I can’t get anywhere. I’ve expanding both sides and seeing if that gets me anywhere but I don’t see how it […]

Learning how to flip equations

I took Algebra and Geometry in high school, never thought I’d use them, then became a programmer. I guess I was wrong. To date, I have the hardest time taking equations and “flipping them,” ie: rewriting an equation to find the reverse. For the current problem I’m dealing with, I have a formula that converts […]

Cubic trig equation

I’m trying to solve the following trig equation: $\cos^3(x)-\sin^3(x)=1$ I set up the substitutions $a=\cos(x)$ and $b=\sin(x)$ and, playing with trig identities, got as far as $a^3+a^2b-b-1=0$, but not sure how to continue. Is there a way to factor this so I can use the zero product property to solve? Thanks for any help/guidance! P

Evaluating the sum $1\cdot 10^1 + 2\cdot 10^2 + 3\cdot 10^3 + \dots + n\cdot 10^n$

How can I calculate $$1\cdot 10^1 + 2\cdot 10^2 + 3\cdot 10^3 + 4\cdot 10^4+\dots + n\cdot 10^n$$ as a expression, with a proof so I could actually understand it if possible?

Finding a polynomial with a given shape

As a followup to $\qquad$Equation of a curve I’ll ask the following question . . . Does there exist a polynomial $f(x)$ with real coefficients such that $f(0) = 2,\;\;f(1) = 3,\;\;f(2)=0,\;\;f(3)=1$. $f$ is$\;$increasing on $(-\infty,1],\;$decreasing on $[1,2],\;$ increasing on $[2,\infty)$. $f$ has exactly one inflection point. Assuming such a polynomial exists, Can someone provide […]

What is the the $n$ times iteration of $f(x)=\frac{x}{\sqrt{1+x^2}}$?

We were asked to determine the composition $f \circ f \circ f \circ…\circ f $, $n$ times, where $f(x)=\dfrac{x}{\sqrt{1+x^2}}$. Anyone has an idea?