# Homework Help

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• Reference
The field of the text is its topic; in this case, the topic is the ceremony to congratulate the seniors who have achieved high results because of their hard work. The mode is both written and oral,...

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• Reference
If you address your colleagues at a youth club meeting, you would use more formal language, meaning you would not likely use slang or incomplete sentences. Instead, if you are chairing the meeting,...

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• Grammar and speech
It can sometimes be difficult to discern the differences between informal and formal language, which so often comes down to a single case of word choice or order, small syntactical adjustments,...

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• Law and Politics
Thomas Jefferson wrote the Declaration of Independence. In the Declaration of Independence, there are many ideas about the role of government and about the rights the people should have. A big part...

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• Death, be not proud
"Death, be not proud" is a sonnet by the iconic English metaphysical poet John Donne. The fourteen-line poem, which is also referred to as "Holy Sonnet X,"deals with the fear of death, overcoming...

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• Grammar
The primary reason this sentence is ambiguous is because of the unclear placement and use of the word "right." In the English language, the word "right" can mean two very different things. The...

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• Reference
'Ambiguity' means a lack of clarity or concrete detail, which can make it difficult to discern meaning. In the case of these two sentences, the lack of context and detail make it possible to...

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• Math
Given \frac{dy}{dx}=\frac{1}{\sqrt{4-x^2}}, y(0)=\pi , we have to find y. So we can write, dy=\frac{dx}{\sqrt{4-x^2}} Integrating both sides we have, y=\int \frac{dx}{\sqrt{4-x^2}}+C Now...

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• Math
Complete the square at the denominator: x^4 + 2x^2 + 2 = (x^2)^2 + 2x^2 + 1 + 1 = (x^2 + 1)^2 + 1. Now we see the substitution y = x^2 + 1, then dy = 2x dx, and the integral becomes int...

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• Math
By completing the square and making simple substitution, we will reduce this integral to a table one. -x^2-4x = -(x^2 + 4x + 4) + 4 = -(x+2)^2 + 4 = 4 - (x+2)^2. Now make a substitution y =...

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• Math
Recall that (arccos(x))' = -1/sqrt(1-x^2) and make the substitution y = arccos(x), then dy = -1/sqrt(1-x^2). The limits of integration are from arccos(0) = pi/2 to arccos(1/sqrt(2)) =...

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• Math
We have to evaluate the definite integral: \int_{0}^{1/\sqrt{2}}\frac{arc sinx}{\sqrt{1-x^2}}dx Let arc sinx=t Differentiating both sides we get, \frac{1}{\sqrt{1-x^2}}dx=dt...

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• Math
Make the substitution u = sin(x), then du = cos(x) dx. The integration limits for u are from sin(0) = 0 to sin(pi/2) = 1, and the integral becomes int_0^1 (du)/(1 + u^2) = arctan(1) -...

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• Math
Make the substitution y = e^(-x), then dy = -e^(-x) dx and e^(-2x) = y^2. The limits of integrations for y become from e^(-ln2) = 1/e^(ln2) = 1/2 to e^(-ln4) = 1/e^(ln4) = 1/4. The...

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• Math
Make the substitution u = sqrt(4x^2 - 9), then du = (4x)/sqrt(4x^2 - 9) dx. Inversely, dx =sqrt(4x^2 - 9)/(4x) du = u/(4x) du and 4x^2 = u^2 + 9. The limits of integration become from...

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• Math
Make a substitution y = 2x, then dy = 2 dx and the limits of integration for y are from 0 to 3. The integral becomes the table one: int_0^3 1/(1+y^2) (dy)/2 = 1/2 (tan)^(-1)(y) |_(y=0)^3...

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• Math
We have to evaluate the integral : \int_{0}^{\sqrt{2}}\frac{dx}{\sqrt{4-x^2}} let x=2sin t So, dx=2cos t dt When x=0, t=0 x=\sqrt{2}, t=\pi/4 So we have,...

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• Math
Make a substitution x = 3y, then dx = 3 dy and the limits of integration for y are from 0 to 1/18. The integral becomes a table one: int_0^(1/18) (9 dy)/(sqrt(9-9y^2)) = int_0^(1/18) (3...

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• Math
We have to evaluate the integral : \int \frac{sec^2x}{\sqrt{25-tan^2x}}dx Let tanx =t So, sec^2x dx=dt Therefore we have, \int \frac{sec^2x}{\sqrt{25-tan^2x}}dx=\int...

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• Math
Recall that the indefinite integral is denoted as: int f(x) dx =F(x)+C There properties and basic formulas of integration we can apply to simply certain function. For the problem int...

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• Math
We have to evaluate \int \frac{dx}{\sqrt{1-4x^2}} Let x=\frac{1}{2} sint  So, dx= \frac{1}{2}cost dt Hence we have, \int \frac{dx}{\sqrt{1-4x^2}}=\int \frac{\frac{1}{2}cost...

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• Math
We have to evaluate the integral: \int \frac{dx}{\sqrt{9-x^2}} let x=3sint So, dx=3cost dt Hence we have, \int \frac{dx}{\sqrt{9-x^2}}=\int \frac{3cost}{\sqrt{9-9sin^2t}}dt...

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• Math
First, check that the given point satisfies the equation: arctan(1 + 0) = 0 + pi/4 is true. The slope of the tangent line is y'(x) at the given point. Differentiate the equation with respect to...

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• Math
Recall that the derivative of a function f at a point x is denoted as f'(x). The given function: f(x)= arcsin(x)+arccos(x) has inverse trigonometric terms. We can solve for the derivative of...

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• Math
We use the product rule, (uv)' = u'v + uv', for u = x^2 and v = arctan(5x), and then the chain rule: h'(x) = 2x*arctan(5x) + x^2 (arctan(5x))' = = 2x*arctan(5x) + (5 x^2)/(1 + 25x^2).

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• Math
The given function f(x) = arctan(sqrt(x)) is in a inverse trigonometric form. The basic derivative formula for inverse tangent is: d/(dx) arctan(u) = ((du)/(dx))/sqrt(1-u^2) . Using...

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• Math
The derivative of a function with respect to t is denoted as f'(t). The given function: f(x) = arcsin(t^2)  is in a form of a inverse trigonometric function. Using table of derivatives, we have...

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• Math
The range of arcsine function is [-pi/2, pi/2], 1/2 belongs to this interval and sine is monotone on this interval. Therefore we may apply sine for both parts and don't lose any solutions or get...

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• Math
arccos(1) Let this expression be equal to y. y = arccos(1) Rewriting it in terms of cosine function, the equation becomes: cos(y) = 1 Base on the Unit Circle Chart, cosine is 1 at angles 0...

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• Math
arccos(1/2) Let this expression be equal to y. y = arccos(1/2) Rewriting this in terms of cosine function the equation becomes: cos(y) =1/2 Base on the Unit Circle Chart, cosine is 1/2 at...

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• Math
arcsin(0) Let this expression be equal to y. y =arcsin(0) Re-writing this equation in terms of sine function, it becomes: sin (y) = 0 Base on the Unit Circle Chart (see attached figure), sine...

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• Math
arcsin(1/2) Let this expression be equal to y. y =arcsin(1/2) Rewriting it in terms of sine function, the equation becomes: sin(y) = 1/2 Base on the Unit Circle Chart (see attached figure),...

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• Math
To evaluate the integral: int_(-4)^(4) 3^(x/4) dx , we follow the formula based from the First Fundamental Theorem of Calculus: int_a^bf(x)dx=F(b)- F(a) wherein f is a continuous and F is...

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• Math
Recall the First Fundamental Theorem of Calculus: If f is continuous on closed interval [a,b], we follow: int_a^bf(x)dx = F(b) - F(a) where F is the anti-derivative of f on [a,b]. This...

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• Math
By definition, if the function F(x) is the antiderivative of f(x) then we follow the indefinite integral as int f(x) dx = F(x)+C where: f(x) as the integrand F(x) as the...

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• Math
Indefinite integral are written in the form of int f(x) dx = F(x) +C where: f(x) as the integrand F(x) as the anti-derivative function C as the arbitrary constant known...

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• Math
To use logarithmic differentiation, take natural logarithm of both sides: ln(y) = (x-1)ln(x). Then differentiate this equation with respect to x and obtain (y')/y = (x-1)/x + ln(x) = 1 - 1/x +...

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• Math
The point is really on the graph, because 2^(-(-1)) = 2. The tangent line has the slope of y'(-1), and the equation y - 2 = y'(-1)(x+1). It is clear that y'(x) = -2^(-x) ln2 and y'(-1) =...

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• Math
The expression under the logarithm (t/t) appears to be constant (1), therefore the entire function is a constant, and its derivative is zero.

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• Math
Derivative of a function h with respect to t is denoted as h'(t). The given function: h(t) = log_5(4-t)^2 is in a form of a logarithmic function. From the derivative for logarithmic functions,...

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• Math
y= log_3(x^2-3x) The derivative formula of a logarithm is: d/(dx) [log_a (u)] = 1/(ln(a) * u) * (du)/(dx) Applying this formula, the derivative of the function will be: (dy)/(dx) = d/(dx)...

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• Math
y=log_4(5x + 1) The derivative formula of a logarithm is: d/(dx) [log_a (u)] = 1/(ln(a) * u) * (du)/(dx) Applying that formula, the derivative of the function will be: (dy)/(dx) = d/(dx)[...

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• Math
f(t) = 3^(2t)/t To take the derivative of this function, use the quotient rule (u/v)'= (v*u' - u*v')/v^2. Applying that, f'(t) will be: f'(t) = (t * (3^(2t))' - 3^(2t)*(t)')/t^2 f'(t) =...

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• Math
The derivative of a function f at a point x is denoted as y' = f'(x). There are basic properties and formula we can apply to simplify a function such as the Product Rule provides the formula:...

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• Math
Recall that the derivative of a function f at a point x is denoted as y' = f'(x) . There basic properties and formula we can apply to simplify a function. For the problem y = x(6^(-2x)), we...

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• Math
The given function: f(x)=x9^x  has two factors since that is the same as f(x) =x* 9^x . In this form, we can apply the Product Rule for derivative. Product Rule provides the formula: f(x) =...

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• Math
y=6^(3x-4) The derivative formula of an exponential function is: d/(dx) (a^u) = ln(a) * a^u * (du)/dx Applying this formula, the derivative of a function will be: (dy)/(dx) = d/(dx)...

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• Math
y=5^(-4x) The derivative formula of an exponential function is: d/(dx) (a^u) = ln(a) * a^u * (du)/(dx) Applying this formula, the derivative of the function is: (dy)/(dx) = d/(dx)(5^(-4x))...

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• Math
The derivative of f(x) with respect to x is denoted a f'(x). The given function f(x) = 3^(4x) is in exponential form which means we can apply the basic integration formula: d/(dx)(a^u) =...

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By definition, the derivative of f(x) with respect to x is denoted a f'(x) where f'(x) = lim (f(x+h) -f(x))/h  as h->0` . Instead of using the limit of difference quotient, we may apply the...