For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.

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4answers
87 views

If $A+B=\pi/3$ then what will maximum value of $\tan(A).\tan(B)$?

Suppose I am given that $$A+B=\frac{\pi}{3}$$ then what will be maximum value of $$\tan(A).\tan(B)=?$$ $$\tan(A+B)=\frac{\tan(A)+\tan(B)}{1-\tan(A).\tan(B)}=\sqrt{3}$$ then ...
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3answers
122 views

Is it correct to say that $\lim_{x\to\infty}e^x=\infty$?

I saw $$\lim_{x\to\infty}e^x=\infty$$ in a textbook, but I think the limit of the left part doesn't exist. So left part doesn't equal right part. Am I right?
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4answers
129 views

Maximizing the area of a triangle with its vertices on a parabola.

So, here's the question: I have the parabola $y=x^2$. Take the points $A=(-1.5, 2.25)$ and $B=(3, 9)$, and connect them with a straight line. Now, I am trying find out how to take a third point on ...
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3answers
83 views

Integration question on $\int \frac{x}{x^2-10x+50} \, dx$

How would I integrate $$\int \frac{x}{x^2-10x+50} \, dx$$ I am not sure on how to start the problem
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5answers
88 views

How to calculate derivative of $f(x) = \frac{1}{1-2\cos^2x}$?

$$f(x) = \frac{1}{1-2\cos^2x}$$ The result of $f'(x)$ should be equals $$f'(x) = \frac{-4\cos x\sin x}{(1-2\cos^2x)^2}$$ I'm trying to do it in this way but my result is wrong. $$f'(x) = \frac ...
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4answers
159 views

Integral $\int\frac{dx}{x^5+1}$ [closed]

Calculate $\displaystyle\int\dfrac{dx}{x^5+1}$
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5answers
124 views

Prove that $\int_{1}^{a} \frac 1t dt + \int_{1}^{b} \frac 1t dt = \int_{1}^{ab} \frac 1t dt$

Prove that $$\int_{1}^{a} \frac 1t dt + \int_{1}^{b} \frac 1t dt = \int_{1}^{ab} \frac 1t dt$$ Useful facts: $\int_{1}^{a} \frac 1t dt$ can be written as $\int_{b}^{ab} \frac 1t dt$ Every ...
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5answers
99 views

Solving $\int_7^9 \frac{2}{9 + 16x^2}\,dx$ without using trigonometric substitution?

How to evaluate $\int_7^9 \frac{2}{9 + 16x^2}\,dx$ without using trigonometric substitution? I know how to do it with trig substitution, but the problem I'm doing requires me to do it with algebra ...
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6answers
118 views

Does $\int_0^\infty \sin(x^{2/3}) dx$ converges?

My Try: We substitute $y = x^{2/3}$. Therefore, $x = y^{3/2}$ and $\frac{dx}{dy} = \frac{2}{3}\frac{dy}{y^{1/3}}$ Hence, the integral after substitution is: $$ \frac{3}{2} \int_0^\infty ...
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5answers
818 views

prove that the product of three numbers is the greatest if the numbers are equal [closed]

How could we prove that if the sum of three numbers is a constant (for example 30), then the product of the numbers would have its maximum value when the numbers are equal?
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2answers
128 views

to evaluate using L'hopital's rule [closed]

Can anyone please guide me how to use L'Hop rule to evaluate this $$ f(x)=\lim_{t\to 0} \frac{1}{2t} \int_{x-t}^{x+t} s f'(s) ds $$
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3answers
142 views

limit is also bounded if sequence is bounded and converge?

assume $\lim _{n\to \infty }\left(a_n\right)=\:L$,and for every n, $a_n\in \left[c,d\right]$. It is true to say that $L\in \left[c,d\right]$? I can't think of opposite example for this so i think its ...
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3answers
101 views

Evaluate $\displaystyle \int_{0}^{\frac{\pi}{2}} \frac{\sin^2 nx}{\sin^2 x} \text{d}x$ [duplicate]

Evaluate $$ \int_{0}^{\frac{\pi}{2}} \frac{\sin^2 nx}{\sin^2 x} \text{d}x$$ where $n\in\mathbb{N}$ This one is another intriguing question from my worksheet. I'm only allowed to use ...
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2answers
112 views

Evaluating $\int\frac{4x^3-3x^2+6x-27}{x^4+9x^2}dx$

$$\int\frac{4x^3-3x^2+6x-27}{x^4+9x^2}dx$$ this integral get very messy. Can I get a step by step breakdown of solving?
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3answers
295 views

Optimization problem?

Hi I was having trouble figuring out this question. Find the point on the circle $x^2 + y^2 = 1$ in the first quadrant where the tangent line to the circle encloses with the coordinate axes a ...
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4answers
166 views

Evaluate $ \lim_{n\to \infty} \sum_{r=1}^{n}\frac{1}{n+r}$ by expressing as Riemann sum [duplicate]

$$ \lim_{n\to \infty} \left(\frac{1}{n+1} + \frac{1}{n+2} + ..+ \frac{1}{2n}\right)$$ How do i find the limit by expressing it as a definite integral of an appropriate function via Riemann Sums? I ...
1
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2answers
114 views

How to integrate $\int \frac{\sqrt{x}}{x+1}dx$?

How to integrate $$\int \frac{\sqrt{x}}{x+1}dx$$ Can I substitute $x+1$ with $u$?
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3answers
74 views

Is $\nabla\cdot{F} = F\cdot\nabla$?

According to the vector dot product, $a\cdot{}b = b\cdot{a}$ for all $a, b.$ However, is $\nabla\cdot{F} = F\cdot\nabla$ (where $\nabla\cdot{F} = \operatorname{div} F$)?
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3answers
78 views

How to evaluate $ \lim_{x\to -\infty} x \sin \left(\frac{1}{x}\right)\quad?$

$$ \lim_{x\to -\infty} x \sin \left(\frac{1}{x}\right)$$ How is the answer 1??? My attempt As x goes to $-\infty$. -$1/\infty$ is 0, so $\sin (0)=0$ $-\infty \sin (0)= \infty (0)$ Well I don't ...
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4answers
86 views

Find $\lim_{x \to 0^+}x\int_x^1 \frac{\cos t}{t^2}\,dt$

Find $$\lim_{x \to 0^+}x\int_x^1 \dfrac{\cos t}{t^2}\,dt$$ This looks like an interesting problem,but i cannot figure out where to start, can anyone explain
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2answers
76 views

Evaluating $\;\int_{1}^{\ln3}\frac{e^x - e^{2x}}{(1 + e^x)}\,dx$

Find $\int_{1}^{\ln3}(e^x - e^{2x})/(1 + e^x)dx$. I looked through my notes for integration techniques and thought I could try a $u$ substitution but whatever I set $u$ to I can't seem to ...
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3answers
96 views

Evaluate $\int \frac{6x+4}{x^2+4}dx$

Find$$\displaystyle \int \dfrac{6x+4}{x^2+4}dx$$ I'm not really sure where to begin with this one - I know the answer will probably involve an $\arctan$, but I am unsure on how to use $\arctan$ in ...
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3answers
79 views

Help Evaluating $\lim_{x\to\frac{\pi}{2}}\left(\frac{1}{\frac{\pi}{2}-x}-\tan {x}\right)$

Does anyone know how to evaluate the following limit? $$\lim_{x\to\frac{\pi}{2}}\left(\frac{1}{\frac{\pi}{2}-x}-\tan {x}\right)$$ The answer is 0 , but I want to see a step by step solution if ...
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3answers
81 views

verify $\lim_{x\rightarrow0}(4x^2+2x+5)=5$

Verify: $$\lim_{x\rightarrow0}(4x^2+2x+5)=5$$ On a simple linear function it's easy to use the limit definition "$|f(x)-L|$ becomes arbitrarily small" but it won't work in this situation. But I'm ...
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3answers
57 views

Solve $y' = x^4y+x^4y^4$

Solve the differential equation $$y' = x^4y+x^4y^4.$$ I'm not sure how to deal with the $x^4y^4$ term. So far I have only encountered differential equations where the exponent of $y$ was at most one. ...
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2answers
109 views

Evaluation of $\displaystyle \int \sec^3 (x)dx$ [duplicate]

How Can I evaluate $\displaystyle \int \sec^3 (x)dx$ (Without Using Weierstrass Substution or Integration by parts.) $\bf{My\; Try::}$ Let $\displaystyle I = \int\sec^3(x)dx = \int ...
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4answers
69 views

Finding $p'(0)$ for the polynomial of least degree which has a local maximum at $x=1$ and a local minimum at $x=3$

The question goes as follows: Let $p(x)$ be a real polynomial of least degree which has a local maximum at $x=1$ and a local minimum at $x=3$. If $p(1)=6$ and $p(3)=2$, then $p'(0)$ is... What I ...
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3answers
250 views

Summation of Infinite Geometric Series

Determine the sum of the following series: $$\sum_{n=1}^{\infty } \frac{(-3)^{n-1}}{7^{n}} $$ My work: $$\sum_{n=1}^{\infty } \frac{(-3)^{n-1}}{7^{n}} = \sum_{n=1}^{\infty } \frac{-1}{7} ...
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3answers
195 views

Convergence of the series $\sum_{i=1}^\infty \sqrt{2n+1}/n^2$

How does the series $\sum_{i=1}^\infty \sqrt{2n+1}/n^2$ converge? I have yet to receive a result that is not inconclusive. If you could tell me what test you used to confirm its convergence that ...
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4answers
591 views

Why does the difference between the left-hand and right-hand sum get smaller with more subdivisions?

Q) For a given function on a given interval, the difference between the left-hand sum and right-hand sum always gets smaller as the number of subdivisions gets larger. I remember that the answer is ...
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2answers
468 views

solve $\tan(x) = \sqrt{1-x^2}$

I am not sure if you should be deriving it or converting tan into $\sin(x)/\cos(x)$. Even then, I do not know what to do from there
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3answers
404 views

Help using Fundemental Theorem of Calculus for the first time

So I want to try using the fundamental theorem of calculus: $F(x) = \int_a^x f(t) dt$ defined in $[a,b]$ $F'(x) = f(x)$ I think that I have a suitable question: Find $F'(x)$, where ...
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3answers
100 views

what is $\int_{0}^{\infty}e^{-x^2}dx$ [duplicate]

I have a question: $$\int_{0}^{\infty}e^{-x^2}dx=?$$ Thanks for your help. Thanks ahead.
1
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2answers
53 views

Show a double-sided infinite integral of $\sin(x+b)$ exists iff $b=n\pi$

More formally: Show that $$\lim_{a\rightarrow \infty} \int_{-a}^a \sin(x+b)$$ exists if and only if $b=n\pi$ for some $n \in \mathbb{Z}$. I get the intuition fine. The function is just a horizontal ...
1
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5answers
78 views

Calculate $\displaystyle \lim_{x \to \frac{3}{2}} \frac{2x^2-3x}{|2x-3|}$

I know that $\displaystyle \lim_{x \to \frac{3}{2}} \frac{2x^2-3x}{|2x-3|}$ does not exist, because the lateral limits are different and I also know that the absolute value on the denominator has ...
1
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4answers
126 views

Double integral for $\int_{0}^{1} \int_{-1}^{0} \frac {xy}{x^2 + y^2 + 1}\ dy\ dx$

I'm trying to evaluate this $$\int_{0}^{1} \int_{-1}^{0} \frac {xy}{x^2 + y^2 + 1}\ dy\ dx$$ tried substition $$ u = {(x^2+y^2+1)}^{-1} \ \ du = \ln {(x^2+y^2+1)}$$ but du is not found in the ...
1
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2answers
79 views

Convergent or divergent? $\sum_{n=1}^{\infty}{\frac{n^n}{(n+1)^{n+1}}}$

Does the series $$\sum_{n=1}^{\infty}{\frac{n^n}{(n+1)^{n+1}}}$$ converge?
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2answers
119 views

Show that each of the series converges.Help Please

Show that each of the series converges on their respective domains. a) $$\sum_{n=1}^\infty \frac{1}{(1 + nx)^2}, x \in (0,\infty)$$ b)$$\sum_{n=1}^\infty e^{-nx}, x \in(0,\infty)$$ For the first ...
1
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4answers
128 views

Find the limit as $x$ tends towards $\frac{\pi}{4}$

In looking at the corresponding graph and differentiating it after reducing it to a different form, I know the that limit is equal to $2$ but I am unsure as to how I can show this algebraically. Any ...
1
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5answers
117 views

Anyone have a good proof for the second part of FTC?

Does anyone have a good proof for the second part of the fundamental theorem of calculus? I haven't been able to find any good videos on it so far so I'd like someone to write it down and I can throw ...
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3answers
255 views

Derivatives of sine and cosine at $x=0$ give all values of $\frac{d}{dx}\sin x$ and $\frac{d}{dx}\cos x$?

In video 3 of the video lectures by MIT on Single Variable Calculus presented by David Jerison, the latter says: Remarks: $\dfrac{d}{dx}\cos x\left|\right._{x=0}=\lim\limits_{\Delta ...
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4answers
77 views

Given $(a_{n+1} - a_n) \rightarrow g$ show $\frac{a_n}{n} \rightarrow g$

So yea, basically this is the problem. $(a_{n+1} - a_n) \rightarrow g$ show $\frac{a_n}{n} \rightarrow g$ It looks like Cauchy's sequence, but I'm not sure. Can we say that if $(a_{n+1} - a_n) ...
1
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4answers
86 views

Evaluate $\lim_{x\to 0} \frac{(1-x)^{1/3}-(1+x)^{1/2}}{x}$

Evaluate the limit $$\lim_{x\to 0} \frac{(1-x)^{1/3}-(1+x)^{1/2}}{x}$$ I know the limit is $-5\over6$ by looking at the graph of the function, but how can I algebraically show that that is the limit?
1
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2answers
203 views

Integral: $I=\int\limits_{0}^{1}\dfrac{x^2dx}{\sqrt{3+2x-x^2}}$

Evaluate: $\displaystyle I=\int\limits_{0}^{1}\dfrac{x^2dx}{\sqrt{3+2x-x^2}}$
1
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3answers
61 views

Finding $y^{\prime}$ of $y=\log_7 e^{8x}$

Find $y^{\prime}$ of $y=\log_7 e^{8x}$ I know that $\dfrac{d}{dx}(e^{8x})=8e^{8x}$, but I am confused on how to work the rest of the problem. Is this correct: $\log_ex=\ln x$ and that ...
1
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4answers
139 views

Help with derivative of $y=x^2\sin^5x+x\cos^{-5}x$

Find $y^{\prime}$ of $y=x^2\sin^5x+x\cos^{-5}x$ My try: $\dfrac{d}{dx}(x^2\sin^5x)=x^2(-5\sin^4x)+(2x\sin^5x)$ $\dfrac{d}{dx}(x\cos^{-5}x)=x(-5\cos^{-6}x)+1(\cos^{-5}x)$ This doesn't seem ...
1
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3answers
108 views

Nspire CAS spitting out a wrong answer?

Consider the integral: $\int \frac{8x+11}{(2x+3)(x+1)}$ My Nspire CAS tells me that the answer to this is $ln\left((x+1)^3 \cdot (2x+3)\right)$ (replace parentheses with absolute value signs, I ...
1
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2answers
115 views

How to check if the series $\sum_{n=0}^{\infty} \sqrt{n+1}-\sqrt{n}$ is convergent

or divergent?? I tried few tests, but I didn't success to discover if the series is convergent or is divergent... $$\sum_{n=0}^{\infty} \sqrt{n+1}-\sqrt{n}$$ Thank you!
1
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6answers
551 views

How do you derive this trig identity from the common ones? $\cos^2x=\frac{1+\cos2x}{2}$

$$\cos^2x=\frac{1+\cos2x}{2}$$ Just came across this identity one today. Where does this come from? Is this an easy derivation from the more popular identities, or is this one you just take it at ...
1
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3answers
164 views

Find $\lim_{ n \to \infty} (\frac{n!}{n^n})^{\frac{1}{n}}$

Find the limit : $\lim_{ n \to \infty} (\frac{n!}{n^n})^{\frac{1}{n}}$ My working : Let $$t = \lim_{ n \to \infty} (\frac{n!}{n^n})^{\frac{1}{n}}$$ Now taking log on both sides : $$\log t = ...