Calculus Homework Answers & Questions

Waiting for answer When would you use u substitution twice?
When we are reversing a differentiation that had the composition of three functions. Here is one example. ##int sin^4(7x)cos(7x)dx## Let ##u=7x##. This makes ##du = 7dx## and our i...
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Waiting for answer How do you find the vertical asymptotes of the function ##y=...
The vertical asymptotes of ##y=(x^2+1)/(3x2x^2)## are ##x=0## and ##x=3/2##. To find the vertical asymptotes we set the denominator equal to zero. ##3x2x^2=x(32x)=0rArrx=0 or x=3/2## Look at the graph below. Sometimes, the denominator is equal to zero at the same xvalue that makes...
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Waiting for answer How do you find the Maclaurin series for ##cos^2 (x)##?
##1x^2+x^4/32/45 x^6+x^8/315+\cdots## There are two methods. 1) Let ##f(x)=cos^2(x)## and use ##f(0)+f'(0)x+(f''(0))/(2!)x^2+(f'''(0))/(3!)x^3+\cdots## We have, by the and/or ,...
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Waiting for answer How do you show that the derivative of an odd function is ev...
I think you could do it like this: Let f(x) be the odd function we're gonna work with. So, f(x) = f(x), By the definition of the derivative: f'(x) = ##lim_(h 0)(f(x+h)  f(x))/h##. We want to prove that f'(x) = f'(x). Now, f'(x) is: ##f'(x) = lim_(h 0)(f(x+h)f(x))/h## . Since f(x) is...
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Waiting for answer What is the integral of ##e^(7x)##?
It's ##1/7e^(7x)## What you want to calculate is: ##int e^(7x)dx## We're going to use . Let ##u = 7x## Differentiate (derivative) both parts: ##du = 7dx## ##(du)/7 = dx## Now we can replace everything in the integral: ##int 1/7 e^u du## Bring the constant upfront ##1/7 int e^u du## The integral o...
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Waiting for answer What is the integral of ##int ( 1 / (25 + x^2) ) dx ##?
##int(1/(25+x^2))dx=1/5 tan^1 (x/5) +C## ##int(1/(25+x^2))dx ## ##dx/d(theta)=5tantheta## ##dx= 5sec^2theta *(d)theta## ##int(1/(25+25tan^2theta))* 5sec^2theta*(d)theta## ##int(...
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Waiting for answer What is the xcoordinate of the point of inflection on the g...
We find the Inflection Points of ##y## by finding the second derivative of the function (y''), and the xvalues at which y'' equals 0. We look for the zeroes because at those points the concavity (or the direction in which the slope of the function ##f(x)## is trending) has leveled off; it is at...
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Waiting for answer How do you find the integral of ##x^3 * sin( x^2 ) dx##?
##I=1/2(x^2cosx^2+sinx^2)+C## ##x^2=t = 2xdx=dt, x^3dx=1/2x^2 2xdx = 1/2tdt## ##int x^3 sinx^2 dx = 1/2 int tsintdt = I## ##u=t = du=dt## ##dv=sintdt = v=int sintdt= cost##...
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Waiting for answer How do you find the third degree Taylor polynomial for ##f(x...
##ln(2)+1/2(x2)1/8(x2)^2+1/24(x2)^3##. The general form of a Taylor expansion centered at ##a## of an analytical function ##f## is ##f(x)=sum_{n=0}^oof^((n))(a)/(n!)(xa)^n##. He...
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Waiting for answer What is the difference between a Tangent line and a secant l...
The to a curve at a given point is a straight line that just "touches" the curve at that point. So if the function is f(x) and if the tangent "touches" its curve at x=c, then the tangent will pass through the point (c,f(c)). The slope of this tangent line is f'(c) ( the derivative of the functio...
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Waiting for answer The functions f and g are defined for all x€R \{ 1,0,1} by...
General approach is to transform both sides of the equality until they start appearing similar. Consider the right hand side, we have: ##(f@g)(x) = f(g(x)) = (1/x +1)/(11/x) =...
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Waiting for answer How to find average acceleration? a bird is flying south at...
##abs(vec a) = 5 m/s^2 ## ## theta = arctan (4/3)## by definition average acceleration over time ##t## is: ##vec a = (Delta vec v)/(t) = 1/t (vec v_2  vec v_1)## using WE as po...
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Waiting for answer How do you find the critical points of a rational function?
To find the critical points of a function, first ensure that the function is differentiable, and then take the derivative. Next, find all values of the function's independent variable for which the derivative is equal to 0, along with those for which the derivative does not exist. These are our crit...
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Waiting for answer What is the derivative of ##mx+b##?
Considering the function (linear): ##y=mx+b## where m and b are real numbers, the derivative, ##y'##, of this function (with respect to x) is: ##y'=m## This function, ##y=mx+b##, represents, graphically, a straight line and the number ##m## represents the SLOPE of the line (or if you want th...
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Waiting for answer Using the integral test, how do you show whether ##sum 1/sqr...
The integral test basically works from the definition of the integral (quick version: the integral is the accumulated sum of infinitely thin differential intervals ##dn## over a specified interval ##a b##). A paraphrased version of the integral test is as follows: Let there be a function ##f(n)...
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Waiting for answer What is the distance between the parallel planes ##3x + y ...
##d = 22/sqrt(26)## Any plane can be represented as ##Pi pp_0, vec n = 0## where ##p = {x,y,x}## is a generic plane point, ##p_0 in Pi## is a plane fixed point and ##vec n...
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Waiting for answer Given ##x^2 + (y – 2)^2 = 4 ## how do you derive a parametri...
The parametric equations are ##x=2costheta## and ##y=2+2sintheta## The equation represents a circle, center ##(0,2)## and radius ##r=2## We use the following parametric equations #...
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Waiting for answer What is the derivative of ##e^(lnx)##?
This is a classic problem in calculus, one that appears to require an exponential derivative but instead can be made incredibly easy using simple algebra and then differentiation. In Algebra II, we learn that because ##e^x## and ##lnx## are inverses, ##e^lna=a## (the e and natural log offset each o...
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Waiting for answer How do I find the equation for a tangent line without deriva...
The equation of the tangent line is the remainder when you divide by (xa)^2. It's EASY for polynomials. You have to examine the analytic geometry of polynomial division and you wil...
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Waiting for answer How do you use a graphing calculator to find the limit of ##...
## lim_(x rarr 0)(12(sqrtx3))/(x9) = 4 ## You can use any graphing program (eg Calculator, Autograph, Internet Sites) in the case I'll use the built in Socratic graphing functiona...
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