Example Solutions for

Step-by-Step Applications of Calculus

Tangent Lines & Planes

Find a line tangent to a curve using the derivative at a point:

tangent line to sin(x) - 1 at x=0
compute the line tangent to (x^3-1)(x+2) at x=1
tangent line of y=x^2 at x=1

Calculate a multivariate tangent plane by examining partial derivatives:

tangent plane to z=x^2+y^2 at x=1,y=1
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Inflection Points

Calculate inflection points step by step:

inflection points x^3 - 3 x^2 + 1
points of inflection t * cbrt(t^2 - 4)
inflection points of 3sin(2x)
inflection pts sgn(x)|x|^(3/2) + x^3 - 3x^2 + 1 over -2 < x < 3
x^3-3x^2-4x+1 inflection points
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Optimization

Explore step by step a variety of tests to discover the extrema:

find the extrema of 3x^4 - 4x^3 - 12x^2 + 1
local extrema of x^3 - 3x + 1

Compute the local maxima or minima of a function:

local minimum for y==x^2-x+1
local maximum x^5 - 10x^3 + 30x

Show the steps for finding the global minima or maxima of a function:

global minima for y==x^4-2x^2+1
minimize sin(x)
global maxima for 1/x
maximize x^5-2x^2+1

Discover how to calculate the extrema of a function on a given interval:

maximize x^5-2x^2+1 on -1<x<1
minimize -2Cos[x] on -1<x<1
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Average Value

Find the average value of an equation on an interval:

average value of 1/x for 1<x<5

Look through the steps to find the average value of a derivative on an interval:

average value of the derivative of Sin(x) for x=1 to 5
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Area between Curves

Learn how to compute the area between curves:

area between 1/2x(x-2)(x+2) and x
area of the region between y=x and y=x^2 for -1/2<x<1
area under 1/x for 1<x<5
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Arc Length

Calculate the arc length of a curve step by step:

arc length of Sin[x] for x between 0 and 2pi

Specify a curve in polar coordinates:

arc length of r=theta^2 for 0<theta<2pi

Specify a curve parametrically:

arc length of {Sin[x], Cos[x], x, x} for 0<x<3pi
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