# y=(x−2)e−xy=(x-2)e-x

**SCIENCE**

**Rosie is building an electromagnet. She has set her electromagnet up with an aluminium core, a…**

Rosie can test her electromagnet’s effectiveness by doing an experiment in which she varies one aspect of her magnet at a time. The ammeter will tell her if she is successfully running electricity…

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**LITERATURE**

**How believable is a story like “The Cold Equations”? What examples from the story make you think…**

Tom Godwin’s short story “The Cold Equations” is realistic and believable. The use of space travel in the story is for colonization, exploration and trade. The consequences of the extra weight of…

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**REMEMBRANCE OF THINGS PAST**

**How important is the role of sleep in Remembrance of Things Past and how does it relate to time…**

Because this collection called Remembrance of Things Past is essentially the philosophical journey of Marcel, his issues with sleep are definitively significant in the stories here. In short, we…

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**SOCIAL SCIENCES**

**Based on what you know about Obama, do you think he is a good president? Why or why not?**

First, you need to decide for yourself whether or not you think Barack Obama is a good president. If you are unsure, then I would suggest doing some quick research on his beliefs and decisions as…

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**AMY LOWELL**

**What is the implied meaning in the poem “A Lady” by Amy Lowell?**

I get a lot of things out this poem. First, the poet is clearly speaking to an older woman, but that woman still has a lot of spunk left. She is “faded / like an old opera tune / Played upon a…

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**HISTORY**

**Which countries did the British Empire rule?**

The British developed an empire that spanned the world. There were many countries that were part of the British Empire. I will highlight some of the countries by continent. In North America, the…

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**HISTORY**

**What was the difference between the Eastern Front and the Western Front during World War I?**

In World War I, there were two major fronts where fighting occurred. These were the Eastern Front and the Western Front. The Germans developed a plan hoping to avoid fighting a two-front war. The…

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**HISTORY**

**What is the connection between specialization and trade?**

There is a connection between specialization and trade. In an ideal economic world free of trade barriers, countries should make the products that they can make more efficiently than countries….

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**HISTORY**

**Why did the Antifederalists fear that the executive branch would become a tyrant?**

The Antifederalists feared that a strong executive branch would lead to a tyrant ruling the country. The Antifederalists didn’t trust a strong federal government because of our experiences while…

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**THE NECKLACE**

**Were her dreams of wanting to live a rich and upper class lifestyle understandable or exaggerated?**

Mathilde’s dreams are both understandable and exaggerated. From an objective standpoint she is not really a sensational beauty. Maupassant makes it clear in the opening sentence that she belongs to…

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**ANIMAL FARM**

**In the novel Animal Farm, how and why was the commandment “No animal shall wear clothes” changed?**

In fact, this commandment was not specifically changed. The others which were changed to suit the needs of the pigs, were: ‘No animal shall sleep in a bed’, which was adjusted to, ‘No animal shall…

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**HISTORY**

**What were the causes of the demobilization at the end of World War I?**

Of course, many nations were involved in the First World War, so this answer will be somewhat broad and general. World War One was at the time the largest and most destructive war in human history….

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**SCIENCE**

**How does pH affect the structure and function of enzymes? ****1 image**

In order to understand how pH affects the structure and function of enzymes, one must first understand what enzymes themselves are. Enzymes are biological molecules that act as catalysts to speed…

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**JULIUS CAESAR**

**Are there any important quotes in Julius Caesar Scene 1 Act 1? I am going to be tested on one.**

When you’re about to be tested on a quote, what you want to do first is ensure you understand the point of the scene itself: what is happening and who is involved. In this case, we have Marullus…

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**DEATH OF A SALESMAN**

**Give one or two examples of a character’s words from Death of a Salesman and discuss why they are…**

Willy Loman is central to Miller’s play and his importance is demonstrated, in part, by the fact that other characters discuss him when he is not on stage. Some of the play’s most poignant…

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**MATH**

**π2–√24≤∫π4π6cos(x)dx≤π3–√24π224≤∫π6π4cos(x)dx≤π324 Use the properties of…**

You need to use the mean value thorem to verify the given inequality, such that: int_a^b f(x)dx = (b-a)*f(c), c in (a,b) Replacing cos x for f(x) and pi/6 for a, pi/4 for b, yields:…

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**MATH**

**2≤∫1−11+x2−−−−−√dx≤22–√2≤∫-111+x2dx≤22 Use the properties of integrals to verify the…**

You need to use the mean value theorem such that: ∫baf(x)dx=(b−a)f(c),c∈(a,b)∫abf(x)dx=(b-a)f(c),c∈(a,b) ∫1−11+x2−−−−−√dx=(1+1)f(c)=2f(c)∫-111+x2dx=(1+1)f(c)=2f(c) You need to verify the monotony of the function

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**MATH**

**∫101+x2−−−−−√dx≤∫101+x−−−−−√dx∫011+x2dx≤∫011+xdx Use the properties of integrals to verify the…**

You need to check if∫101+x2−−−−−√dx≤∫101+x−−−−−√dx∫011+x2dx≤∫011+xdx , using mean value theorem, such that: ∫baf(x)dx=(b−a)f(c),∫abf(x)dx=(b-a)f(c), where c∈(a,b)c∈(a,b)

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**MATH**

**∫40(x2−4x+4)dx≥0∫04(x2-4x+4)dx≥0 Use the properties of integrals to verify the inequality…**

You need to use the mean value thorem to verify the given inequality, such that: ∫baf(x)dx=(b−a)⋅f(c),c∈(a,b)∫abf(x)dx=(b-a)⋅f(c),c∈(a,b) Replacing x2−4x+4×2-4x+4 for f(x)f(x) and 0 for a, 4 for b, yields:

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**MATH**

**∫100|x−5|dx∫010|x-5|dx Evaluate the integral by interpreting it in terms of areas.**

∫100|x−5|dx∫010|x-5|dx To interpret this in terms of area, graph the integrand. The integrand is the function f(x) = |x – 5|. Then, shade the region bounded by f(x) = |x-5| and the x-axis in the…

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**MATH**

**∫2−1|x|dx∫-12|x|dx Evaluate the integral by interpreting it in terms of areas.**

∫2−1|x|dx∫-12|x|dx To interpret this in terms of area, graph the integrand. The integrand is the function f(x) =|x|. Then, shade the region bounded by the graph of f(x)=|x| and the x-axis in the…

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**MATH**

**∫5−5(x−25−x2−−−−−−√)dx∫-55(x-25-x2)dx Evaluate the integral by interpreting it in terms of areas.**

∫5−5(x−25−x2−−−−−−√)dx∫-55(x-25-x2)dx =∫5−5xdx−∫5−525−x2−−−−−−√dx=∫-55xdx-∫-5525-x2dx =I1−I2=I1-I2 I_1 can be be interpreted as area of two triangles;one above the x-axis and the other below axis.Since they are on the…

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**MATH**

**∫0−3(1+9−x2−−−−−√)dx∫-30(1+9-x2)dx Evaluate the integral by interpreting it in terms of areas.**

∫0−3(1+9−x2−−−−−√)dx∫-30(1+9-x2)dx Consider the graph of y=f(x)=1+9−x2−−−−−√1+9-x2 y=1+9−x2−−−−−√y=1+9-x2 y−1=9−x2−−−−−√y-1=9-x2 (y−1)2=9−x2(y-1)2=9-x2 x2+(y−1)2=32×2+(y-1)2=32 This is the equation of circle of radius 3 centred at…

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**MATH**

**∫90((13)x−2)dx∫09((13)x-2)dx Evaluate the integral by interpreting it in terms of areas.**

∫90(13x−2)dx∫09(13x-2)dx To interpret this integral in terms of area, graph the integrand. The integrand is the function f(x)=13x−2f(x)=13x-2 . Then, shade the region bounded by f(x)=13x−2f(x)=13x-2 and the…

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**MATH**

**∫2−1(1−x)dx∫-12(1-x)dx Evaluate the integral by interpreting it in terms of areas.**

You need to evaluate the area enclosed by the curve represented by the function f(x) = (1-x), x axis and the lines x = -1 and x = 2, using the fundamental theorem of calculus, such that:…

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**MATH**

**∫101(x−4ln(x))dx∫110(x-4ln(x))dx Express the integral as a limit of Riemann sums. Do not evaluate…**

∫101(x−4ln(x))dx∫110(x-4ln(x))dx To express this definite integral as limit of Riemann’s Sum, apply the formula: ∫baf(x)dx=limn→∞∑i=1∞f(xi)Δx∫abf(x)dx=limn→∞∑i=1∞f(xi)Δxwhere Δx=b−anΔx=b-an

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**MATH**

**∫62×1+x5dx∫26×1+x5dx Express the integral as a limit of Riemann sums. Do not evaluate the…**

You have to recall the definition of the Reiman Integral ∫baf(x)dx=limn→∞∑i=1nf(x(i))Δx∫abf(x)dx=limn→∞∑i=1nf(x(i))Δx whereΔx=b−anandx(i)=a+iΔxwhereΔx=b-anandx(i)=a+iΔx xx a=2andb=6a=2andb=6

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**MATH**

**∫ba(x2)dx=b3−a33∫ab(x2)dx=b3-a33 Prove that**

You need to evaluate the definite integral, such that: ∫baf(x)dx=F(b)−F(a)∫abf(x)dx=F(b)-F(a) ∫bax2dx=x33∣∣∣ba∫abx2dx=x33∣ab ∫bax2dx=b33−a33∫abx2dx=b33-a33 ∫bax2dx=b3−a33∫abx2dx=b3-a33 Hence,…

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**MATH**

**∫baxdx=b2−a22∫abxdx=b2-a22 Prove that**

You need to use the fundamental theorem of calculus, to prove the equality, such that: ∫baf(x)dx=F(b)−F(a)∫abf(x)dx=F(b)-F(a) You need to replace x for f(x), such that: ∫baxdx=x22∣∣∣ba∫abxdx=x22∣ab

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**MATH**

**∫51(x2)(e−x)dx,n=4∫15(x2)(e-x)dx,n=4 Use the Midpoint Rule with the given value of nn to…**

You need to evaluate the definite integral using the mid point rule, hence, first you need to evaluate Δx:Δx: Δx=b−an⇒Δx=5−14=1Δx=b-an⇒Δx=5-14=1 You need to denote each of the 4…

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**MATH**

**∫20xx+1dx,n=5∫02xx+1dx,n=5 Use the Midpoint Rule with the given value of nn to approximate…**

You need to use the midpoint rule to approximate the interval. First, you need to find ΔxΔx , such that: Δx=b−anΔx=b-an The problem provides b=2, a=0 and n = 5, such that:

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**MATH**

**∫π20cos4(x)dx,n=4∫0π2cos4(x)dx,n=4 Use the Midpoint Rule with the given value of nn to…**

You need to use the midpoint rule to approximate the interval. First, you need to find Δx,Δx, such that: Δx=b−anΔx=b-an The problem provides b=π2b=π2 , a=0 and n = 4, such that:

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**MATH**

**∫80sin(x−−√)dx,n=4∫08sin(x)dx,n=4 Use the Midpoint Rule with the given value of nn to approximate…**

You need to use the midpoint rule to approximate the interval. First, you need to find ΔxΔx , such that: Δx=b−anΔx=b-an The problem provides b=8, a=0 and n = 4, such that:

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**MATH**

a(t)=sin(t)+3cos(t)a(t)=sin(t)+3cos(t) v(t)=−cos(t)+3sin(t)+av(t)=-cos(t)+3sin(t)+a Now,v(0)=2Now,v(0)=2 Thus,2=−cos(0)+3sin(0)+aThus,2=-cos(0)+3sin(0)+a or,2=−1+0+aor,2=-1+0+a or.,a=3or.,a=3 Now,v(t)=−cos(t)+3sin(t)+3Now,v(t)=-cos(t)+3sin(t)+3

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**MATH**

**v(t)=2t−11+t2,s(0)=1v(t)=2t-11+t2,s(0)=1 A particle is moving with the given data. Find the…**

v(t)=2t−11+t2v(t)=2t-11+t2 position of the particle s(t) is given by, s(t)=∫v(t)dts(t)=∫v(t)dt s(t)=∫(2t−11+t2)dts(t)=∫(2t-11+t2)dt s(t)=2(t22)−arctan(t)+Cs(t)=2(t22)-arctan(t)+C , C is constant s(t)=t2−arctan(t)+Cs(t)=t2-arctan(t)+C Now let’s find C ,…

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**MATH**

**f”(x)=2×3+3×2−4x+5,f(0)=2,f(1)=0f′′(x)=2×3+3×2-4x+5,f(0)=2,f(1)=0 Find ff.**

f”(x)=2×3+3×2−4x+5f′′(x)=2×3+3×2-4x+5 or,f'(x)={(12)x4}+x3−2×2+5x+aor,f′(x)={(12)x4}+x3-2×2+5x+a or,f(x)={(110)⋅x5}+(x44)−{(23)x3}+{(52)x2}+ax+bor,f(x)={(110)⋅x5}+(x44)-{(23)x3}+{(52)x2}+ax+bNow,f(0)=2Now,f(0)=2 i.e.2=bi.e.2=b Also,f(1)=0Also,f(1)=0…

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**MATH**

**f”(x)=1−6x+48×2,f(0)=1,f'(0)=2f′′(x)=1-6x+48×2,f(0)=1,f′(0)=2 Find ff.**

f”(x)=1−6x+48x2f′′(x)=1-6x+48×2 f'(x)=x−3×2+16×3+af′(x)=x-3×2+16×3+a Now,f'(0)=2Now,f′(0)=2 i.e.2=ai.e.2=a Thus,f'(x)=x−3×2+16×3+2Thus,f′(x)=x-3×2+16×3+2 f(x)=(x22)−x3+4×4+2x+bf(x)=(x22)-x3+4×4+2x+b Now,f(0)=1Now,f(0)=1 Thus,b=1Thus,b=1…

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**MATH**

**f'(u)=u2+u−−√u,f(1)=3f′(u)=u2+uu,f(1)=3 Find ff.**

You need to evaluate f(u) using the antiderivative of the function f'(u), such that: ∫f'(u)du=f(u)+c∫f′(u)du=f(u)+c ∫u2+u−−√udu=∫u2udu+∫u−−√udu∫u2+uudu=∫u2udu+∫uudu

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**MATH**

**f'(t)=2t−3sin(t),f(0)=5f′(t)=2t-3sin(t),f(0)=5 Find ff.**

f'(t)=2t−3sin(t)f′(t)=2t-3sin(t) f(t)=∫(2t−3sin(t))dtf(t)=∫(2t-3sin(t))dt f(t)=2(t22)−3(−cos(t))+Cf(t)=2(t22)-3(-cos(t))+C ,C is constant f(t)=t2+3cos(t)+Cf(t)=t2+3cos(t)+C Now , evaluate C , given f(0)=5 f(0)=5=02+3cos(0)+Cf(0)=5=02+3cos(0)+C 5=3+C5=3+C C=2C=2…

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**MATH**

**f'(x)=sinh(x)+2cosh(x),f(0)=2f′(x)=sinh(x)+2cosh(x),f(0)=2 Find ff.**

You need to evaluate the function f using the provided information, hence, you need to apply the antiderivative, such that:

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**MATH**

**f'(x)=x3−−√+x2−−√3f′(x)=x3+x23 Find ff.**

f'(x)=x3−−√+x2−−√3f′(x)=x3+x23 f(x)=∫f'(x)dxf(x)=∫f′(x)dx f(x)=∫(x3−−√+x2−−√3)dxf(x)=∫(x3+x23)dx apply the sum rule, f(x)=∫x3−−√dx+∫x2−−√3dxf(x)=∫x3dx+∫x23dx f(x)=x32+132+1+x23+123+1+Cf(x)=x32+132+1+x23+123+1+C , C is…

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**MATH**

**f'(x)=2ex+sec(x)tan(x)f′(x)=2ex+sec(x)tan(x) Find ff.**

f'(x)=2ex+sec(x)tan(x)f′(x)=2ex+sec(x)tan(x) f(x)=2ex+sec(x)f(x)=2ex+sec(x)

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**MATH**

**f'(x)=cos(x)−(1−x2)−12f′(x)=cos(x)-(1-x2)-12 Find ff.**

f'(x)=cos(x)−(1−x2)−12f′(x)=cos(x)-(1-x2)-12 f(x)=sin(x)−sin−1(x)+cf(x)=sin(x)-sin-1(x)+c

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**MATH**

**y=x+ln(x2+1)y=x+ln(x2+1) Sketch the curve by locating max/mins, asymptotes, points of…**

y=f(x)=x+ln(x2+1)y=f(x)=x+ln(x2+1) a) Asymptotes The function has no undefined points , so it has no vertical asymptotes. For Horozontal asymptotes , check if at x→±∞x→±∞ the function behaves as a line…

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**MATH**

**y=(x−2)e−xy=(x-2)e-x Sketch the curve by locating max/mins, asymptotes, points of inflection,…**

y=(x−2)e−xy=(x-2)e-x (I) Asymptotes To determine its horizontal asymptotes, take the limit of this function as x approaches positive and negative infinity. limx→−∞(x−2)e−x=−∞limx→-∞(x-2)e-x=-∞…

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**MATH**

**y=e2x−x2y=e2x-x2 Sketch the curve by locating max/mins, asymptotes, points of inflection,…**

min/max Potential local extrema are found at points where y’=0.y′=0. y’=(2−2x)e2x−x2y′=(2-2x)e2x-x2 (2−2x)e2x−x2=0(2-2x)e2x-x2=0 Since e2x−x2>0e2x-x2>0 we have 2−2x=02-2x=0 x=1x=1 To determine whether the function has…

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**MATH**

**y=sin−1(1x)y=sin-1(1x) Sketch the curve by locating max/mins, asymptotes, points of inflection, etc.**

Here is the graph. We can see that the domain is (−∞,1]U[1,∞)(-∞,1]U[1,∞)We cannot take the arcsin of a number bigger than 1, so therefore if -1<x<1 then we cannot take the arcsin of 1/x. The…

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**MATH**

**y=4x−tan(x),−π2<x<π2y=4x-tan(x),-π2<x<π2 Sketch the curve by locating max/mins,…**

You need to evaluate the asimptotes of the function, such that: lim_(x->-pi/2,x>-pi/2)(4x – tan x) = -2pi – tan(-pi/2) = 2pi + tan (pi/2) = oo lim_(x->pi/2,x<pi/2)(4x – tan x) = 2pi -…

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**MATH**

**y=exsin(x),−π≤x≤πy=exsin(x),-π≤x≤π Sketch the curve by locating max/mins, asymptotes,…**

y=exsin(x),−π≤x≤πy=exsin(x),-π≤x≤π 1) Asymptotes: There is no undefined point, so the function has no vertical asymptote. Since -∞∞ and ∞∞ are not in the domain of the function , so it has no…

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**MATH**

**y=x2+1−−−−−√3y=x2+13 Sketch the curve by locating max/mins, asymptotes, points of…**

y=x2+1−−−−−√3y=x2+13 a) Asymptotes Since the function has no undefined point, so it has no vertical asymptote. For horizontal asymptotes check if at x→±∞→±∞ , the function behaves as a line…

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