guys right now I'm writing finals in my University, so can you help me solve some of the easy math problems? So here is one of them for example
6. We have [math]\displaystyle p(x) = g^2(x)f(x)[/math]. So [math]\displaystyle p'(x) = 2g(x)g'(x)f(x) + g^2(x)f'(x)[/math] which implies [math]\displaystyle p'(-1) = 2g(-1)g'(-1)f(-1) + g^2(-1)f'(-1) = 2(3)(5)(4) + (3)^2(1/3) = 123[/math]
>1.
[eqn]\lim_{x \to 4^-} \frac{x-4}{x^2+x-20} = \lim_{x \to 4^-} \frac{1}{x+5} = \frac{1}{4+5} = \frac{1}{9}[/eqn]
A. 1/9
>2.
[eqn]\lim_{x \to \pi^+} \frac{x\cos(x)}{\sin(x)} = \lim_{x \to \pi^+} x\cos(x) \times \lim_{x \to \pi^+} \frac{1}{\sin(x)} = \pi\cos(pi) \times \frac{1}{\sin(\pi)} = -\pi \times -\infty = \infty[/eqn]
D. [eqn]\infty[/eqn]
>3.
[eqn]\lim_{x \to 2} \frac{\ln{(x^3-7)}}{4-x^2} = \lim_{x \to 2} \frac{\frac{3x^2}{x^3-7}}{-2x} = \lim_{x \to 2} \frac{-3x}{2(x^3-7)} = \frac{-3 \times 2}{2(2^3-7)} = -3[/eqn]
C. -3
>4.
[eqn]\lim_{x \to \infty} \frac{x^2}{4-x} + x = \lim_{x \to \infty} \frac{-4x}{x-4} = -4[/eqn]
D. -4
>5.
[eqn]f(x) = \frac{e^{2x}}{5+e^{4x}}\\
f'(x) = \frac{(5+e^{4x})2e^{2x} - (e^{2x}4e^{4x})}{(5+e^{4x})^2}\\
f'(x) = \frac{10e^{2x} - 2e^{6x}}{(5+e^{4x})^2}\\
f'(0) = \frac{10e^{0} - 2e^{0}}{(5+e^{0})^2}\\
f'(0) = \frac{10 - 2}{(5+1)^2}\\
f'(0) = \frac{8}{36}\\
f'(0) = \frac{2}{9}[/eqn]
B. 2/9
>6.
[eqn]p(x) = g^2(x)f(x)\\
g(-1) = 3, g'(-1) = 5,f(-1) = 4, f'(-1) = \frac{1}{3}\\
p'(-1) = f'(-1)g^2(-1) + 2g(-1)g'(-1)f(-1)\\
p'(-1) = (\frac{1}{3} \times 9) + (2 \times 3 \times 5 \times 4)\\
p'(-1) = 3 + 5!\\
p'(-1) = 3 + 120\\
p'(-1) = 123[/eqn]
D. 123
The only reason why I did this is because I want to practice my Calculus as well. This shit was extremely easy, though.
multiple choice maths exams what in the fug
>>7722137
what kind of backwater swamp of a university do you attend?
or you sure its not TAFE/community college?
>>7722137
Holy shits are these even called Exams?
I fucking hate limits and all that is associated with them
I still don't get the point of derivatives
Why am I so retarded
>>7724367
Yeah
>>7724367
Derivatives are useful for tons of things. One common example is for approximation. Let's say you want to approximate (estimate) sqrt(3), but you don't have a calculator or anything on you. You could use derivatives for that:
[eqn]f(4-1)\approx (f'(4)\times-1)+f(4)\\
f(3)\approx f(4)-f'(4)\\
f(3)\approx 2-\frac{1}{4}\\
f(3)\approx \frac{7}{4}[/eqn]
You can also use derivatives to graph a function by hand (when you're given only the function, a pencil, and paper; no graphing calculator whatsoever). Derivatives can be used to find the velocity, acceleration, jerk, etc. of an object. Many basic physics formulas are actually from derivatives, such as ΔX = Vt + 0.5*a*t^2 is actually found and proven through derivatives, etc.
Meanwhile, we can use limits for a variety of different reasons. Limits can also be used to find derivatives; in fact, the linear approximation equation stems from the limit definition of the derivative of a function. Limits can be used to find antiderivatives as well, and we can use limits to find the area under a curve (aka definite integrals. It might seem petty, but to give you an example, in Statistics, the probability(s) in Normal Distributions are approximated through definite integrals.
E.g. For a normal distribution with μ = 0 and σ = 1, p(x <= -1):
[eqn]p(x \leq -1)=\int_{-\infty}^{-1} \frac{e^{-\frac{x^2}{2}}}{\sqrt{2\pi}}\\
p(x \leq -1) \approx 0.158655...[/eqn]
Yep. Limits can literally be used to find the probability of event(s) happening. Limits can also be used to find indefinite integrals (a.k.a. no bounds). There are a variety of other uses for integrals and derivatives that I am probably missing out on, but I hope you get the point.
They're important.
>>7725627
That second Tex equation came out a bit funky... let me try again (and if this doesn't work then I give up. Just plug it into a Tex editor to view if you're curious):
[eqn]
p(x \leq -1) = \int_{-\infty}^{-1} \frac{e^{-\frac{x^2}{2}}}{\sqrt{2\pi}}\\
p(x \leq -1) \approx 0.158655...
[/eqn]
A lot of this stuff doesn't seem too different from trig. I thought math was supposed to get more fun after trig.
Why did your teacher put brackets on #4 OP? It seems a bit redundant and is pissing me off, frankly.
>>7725853
because there are two terms in that one, so we want to make sure the "+x" is included in the limit calculation
>>7722137
you are writing the exams? so you are the professor? and you don't know calculus?
remind me not to go to your school!
>>7722137
R u srs
>>7725853
That's proper notation
This is a bait, you are solving his exam.
