Calculus Homework Answers & Questions

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Answered Find atleast two equations of a line through ( 4,
Find atleast two equations of a line through ( 4, 13 ) that is tangent to y = 2x^2  1
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Waiting for answer Find the local maximum and minimum values of ##f## using bot...
## f(5)=1/10## is a local max. ## f(5)=1/10## is a local min. ##f(x)={x}/{x^2+25}## By , ##f'(x)={1 cdot(x^2+25)x cdot 2x}/{(x^2+25)^2}={25x^2}/{(x^2+25)^2}=0## ##= x= pm5" "...
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Waiting for answer What is the derivative of ##1+tan^2x##?
##2tanxsec^2x## differentiate using the##color(blue)(" chain rule ") ## rewrite ## tan^2x = (tanx)^2## ##d/dx[ 1 + (tanx)^2] = 2(tanx) d/dx(tanx) ## ## = 2tanxsec^2x ##
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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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