Calculus Homework Answers & Questions

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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Waiting for answer How do you find the derivative of ##y=cos(x)## from first p...
Using the definition of a derivative: ##dy/dx = lim_(h 0) (f(x+h)f(x))/h##, where ##h = deltax## We substitute in our function to get: ##lim_(h 0) (cos(x+h)cos(x))/h## Using the Trig identity: ##cos(a+b) = cosacosb  sinasinb##, we get: ##lim_(h 0) ((cosxcos h  sinxsin h)cosx)/h...
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Waiting for answer How do you use the epsilon delta definition to prove that th...
Please see below. The preliminary analysis is a bit long. If you just want to read the proof, scroll down. Preliminary analysis We want to show that ##lim_(xrarr2)(x^27x+3) = 7...
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Waiting for answer What is a corner in calculus?
A derivative at a specified point is only defined for a function where there is only one slope at that specified point. A corner is one type of shape to a graph that has a different slope on either side. It is similar to a cusp . You may see corners in the context of absolute value functions, like...
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Waiting for answer How do you integrate ##ln(x^2x+2)dx##?
##=(x 1/2) ln(x^2x+2)  2x + sqrt 7 arctan ( (2x1)/sqrt 7)## ##int dx qquad ln(x^2x+2)## we use IBP ##int u v' = uv  int u' v## and the trick here is ##u = ln(x^2x+2),...
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Waiting for answer If a 200 pound cable is 100 feet long and hangs vertically f...
Since the density of the cable is ##{200}/{100}=2## lb/ft, the small piece of the cable of length ##dx## pulls down with the force ##2dx## lb. Let ##x## be the distance of the piece of the cable from the top of the building. To lift this piece up to the top, it requires the force of ##dW=2d...
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Waiting for answer Find the volume of the largest rectangular box in the ﬁrst...
By solving for ##z##, ##x+2y+3z=6 Leftrightarrow z= 21/3x2/3y## Let ##(x,y,z)## be the vertex on the plane, where ##x,y,z 0##. The volume ##V## of the rectangular box can be expressed as ##V=xyz=xy(21/3x2/3y)=2xy1/3x^2y2/3xy^2## Second Partial Test Let us find critical points. ##V_x=2...
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Waiting for answer What is the derivative of ##sinx(sinx+cosx)##?
The answer is ##cos^2(x)sin^2(x)+2cos(x)sin(x)=cos(2x)+sin(2x)## First, use the to say ##d/dx(sin(x)(sin(x)+cos(x)))=cos(x)(sin(x)+cos(x))+sin(x)(cos(x)sin(x))## Next, expand t...
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Waiting for answer What is the antiderivative of ##3e^x##?
##3e^x+C## You should already know that the derivative of ##e^x## is just ##e^x##. Also, when differentiating, multiplicative constants remain and are not altered. Since the two com...
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Waiting for answer Suppose that we don't have a formula for g(x) but we know th...
Linearization of f(x) at a ##L(x)=f(a)+f'(a)(xa)## Let us find the linearization ##L(x)## of ##g(x)## at ##1##. Since ##g(1)=3## and ##g'(1)=sqrt{(1)^2+8}=3##, ##L(x)=g(1)+g'(1)(x1)=3+3(x1)=3x6##. So, we can approximate: ##g(0.9) approx L(0.9)=3(0.9)6=3.3## ##g(1.1) approx L(1.1)=3(...
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Waiting for answer What is the line of symmetry of the graph of ##y=1/(x1)##?
The graph is a hyperbola, so there are two lines of symmetry: ##y=x1## and ##y=x+1## The graph of ##y = 1/(x1)## is a hyperbola. Hyperbolas have two lines of symmetry. both li...
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Waiting for answer How do you remove a removable discontinuity?
You can remove a removable discontinuity by redefining the function at the "problem spot". Removable discontinuities are where the graph of a function has a hole. This will occur in rational functions at an xvalue that makes both the denominator and the numerator equal zero. For example, the grap...
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Waiting for answer How do you know when to use L'hospital's rule twice?
L'hospital's rule is used when an initial evaluation of a limit results in an indeterminate form such as ##0/0## or ##(+oo)/(+oo)## If, after application of L'hospital's rule, your limit evaluation produces another indeterminate form you apply L'ospital's rule again. L'hospital's rule st...
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