Examples of Curve Sketching

Key Questions

• How do you sketch the curve `f(x)=e^x/(1+e^x)` ?

  Information from `f(x)`

  `f(0)=1/{1+1}=1/2 Rightarrow` y-intercept: `1/2`

  `f(x) > 0 Rightarrow` x-intercept: none

  `lim_{x to infty}e^x/{1+e^x}=1 Rightarrow` H.A.: `y=1`

  `lim_{x to -infty}e^x/{1+e^x}=0 Rightarrow` H.A.: `x=0`

  So far we have the y-intercept (in blue) and H.A.'s (in green):

  Information from `f'(x)`

  `f'(x)={e^xcdot(1+e^x)-e^xcdot e^x}/{(1+e^x)^2}=e^x/(1+e^x)^2>0`

  `Rightarrow` `f` is always increasing.

  Information from `f''(x)`

  `f''(x)={e^x cdot (1+e^x)^2-e^xcdot2(1+e^x)e^x}/{(1+e^x)^4}`

  `={e^x(1+e^x)(1-e^x)}/{(1+e^x)^4}={e^x(1-e^x)}/{(1+e^x)^3}`

  `f''(x)>0` on `(-infty,0)` and `f''(x)<0` on `(0, infty)`

  `f` is concave upward on `(-infty,0)` and downward on `(0, infty)`.

  Hence, we have the graph of `f` (in blue):

  Wataru · 1 · Sep 25 2014

• How do you sketch the curve `f(x)=x+sqrt(1-x)` ?
• How do you sketch the curve `f(x)=e^x/(1+e^x)` ?
• How do you sketch the graph of `f(θ)=2cosθ+cos2 θ` for `0≤x≤ 2π` using the first and second derivative?
• How do you graph `y = 2x^3 - 4x + 1` using the first and second derivative?
• How do you graph `F(x,y)=sqrt(x^2+y^2-1)+ln(4-x^2-y^2)`?
• How do you sketch `f(x,y)=arcsin(x^2+y^2-2)`?
• How do you graph `f(x,y) = 10 - 4x - 4y`?
• How do you graph of the function `y=arctan(x)`?
• How do you sketch the graph of `f(x)=x^3-3x^2`?
• How do you graph `f(x) = xsqrt(4-x^2)`
• How do you graph the function `y=5x-x^2`?
• How do you graph `y=3 + 2/(4−x)`?
• How do you graph `f(x)=x^4-3x^2+2x`?
• How do you graph `f(x)=sinx-sqrt(3)cos x` for x is between [0, 2pi ]?
• How do you graph `f(x,y)= 10-4x-5y`?
• How do you sketch `f(x,y) = ln(x^2+y^2)`?
• How do you sketch the graph of `y=5x^3-32x^2+29x+30`?
• How do you find the domain and range of `h(x)=ln(x-6)`?
• How do you graph `x=2y-y^2`?
• How do you sketch the graph of `f(x) = (x^2 - 3x + 9)/(x - 3)`?
• How do you sketch the curve for `y= (x^2+1)/(x^2-4)`?
• How do you sketch the curve of `f(x) = (2x-3) / (2 x -9)^2`?
• How do you use the first and second derivatives to sketch `y = x^3 - 12x - 12`?
• How do you use the first and second derivatives to sketch `y=x^4-2x`?
• How do you use the first and second derivatives to sketch `y = (x+2)(25 -x^2)`?
• How do you use the first and second derivatives to sketch `f(x) = sqrt(4 - x^2)`?
• How do you use the first and second derivatives to sketch `g(x)= ( 1 + x^2 ) / ( 1 - x^2 )`?
• How do you use the first and second derivatives to sketch `y = x / (x^2 - 9)`?
• How do you use the first and second derivatives to sketch `y=(x+2)/(x-1)`?
• How do you use the first and second derivatives to sketch `y=e^x/x`?
• How do you use the first and second derivatives to sketch `y = x - ln |x|`?
• How do you use the first and second derivatives to sketch `y= -(x-2) (x+2) (x-4)`?
• How do you use the first and second derivatives to sketch `f(x)=(x+2)/(x-3)`?
• How do you use the first and second derivatives to sketch `f(x) = (x+1)e^x`?
• How do you use the first and second derivatives to sketch `f(x)=x^2/(x^2+5)`?
• How do you use the first and second derivatives to sketch `f(x) = | (x^2) -1 |`?
• How do you use the first and second derivatives to sketch `f(x)=e^(-x^2)`?
• How do you use the first and second derivatives to sketch `y=(x^3)-(6x^2)+5x+12`?
• How do you use the first and second derivatives to sketch `F(x) = ln(x) / sqrt(x)`?
• How do you use the first and second derivatives to sketch `y = x + (1-x)^(1/2)`?
• How do you use the first and second derivatives to sketch `h(x)=x³-3x+1`?
• How do you use the first and second derivatives to sketch `y = e^(1-2x)`?
• How do you use the first and second derivatives to sketch `-2(x-2)(x+3)(x+4)`?
• How do you use the first and second derivatives to sketch `y=2x^3- 3x^2 -180x`?
• How do you use the first and second derivatives to sketch `y= 3x^4 - 4x^3`?
• How do you use the first and second derivatives to sketch `f(x)= x^4 - 2x^2 +3`?
• How do you graph the derivative of `f(x) = x^2`?
• How do you graph the derivative of `f(x) = x(x-1)`?
• How do you graph the derivative of `f(x) = cos(x)`?
• How do you graph the derivative of `f(x) = log (x)`?
• What is the domain and range of `sqrt((5x+6)/2)`?
• How do you make the graph for `y=ln(1+x/(ln(1-x)))`?
• How do you sketch the graph by determining all relative max and min, inflection points, finding intervals of increasing, decreasing and any asymptotes given `f(x)=x^4-4x^3`?
• How do you sketch the graph by determining all relative max and min, inflection points, finding intervals of increasing, decreasing and any asymptotes given `f(x)=(4x)/(x^2+1)`?
• How do you sketch the graph by determining all relative max and min, inflection points, finding intervals of increasing, decreasing and any asymptotes given `f(x)=(x-4)^2/(x^2-4)`?
• How do you sketch the graph by determining all relative max and min, inflection points, finding intervals of increasing, decreasing and any asymptotes given `f(x)=x-x^(2/3)(5/2-x)`?
• How do you graph `f(x)=6x-15` using the information given by the first derivative?
• How do you graph `f(x)=x^3-3x^2-9x+6` using the information given by the first derivative?
• How do you graph `f(x)=(x^2-4)^2` using the information given by the first derivative?
• How do you graph `f(x)=1/3x^3-6x` using the information given by the first derivative?
• How do you graph `f(x)=1/x-3x^3` using the information given by the first derivative?
• How do you sketch the graph that satisfies f'(x)>0 when x<3, f'(x)<0 when x>3#, and f(3)=5?
• How do you sketch the graph that satisfies f'(x)>0 when -1<x<3, f'(x)<0 when x<-1 and when x>3, and f(-1)=0, f(3)=4?
• How do you sketch the graph that satisfies f'(x)>0 when x does not equal 2, f(2)=1?
• How do you sketch the graph that satisfies f'(x)=1 when x>-2, f'(x)=-1 when x<-2, f(-2)=-4?
• How do you sketch the graph of `1/(x^2-4)`?
• How do you sketch the graph `f(x)=x^3+1`?
• How do you sketch the graph `f(x)=2x^3-12x^2+18x-1`?
• How do you sketch the graph `f(x)=2x^4-26x^2+72`?
• How do you sketch the graph `f(x)=3x^4+2x^3-15x^2+12x-2`?
• Question #bd70e
• How do you sketch the curve `y=x^3-3x^2-9x+5` by finding local maximum, minimum, inflection points, asymptotes, and intercepts?
• How do you sketch the curve `y=e^x-sinx` by finding local maximum, minimum, inflection points, asymptotes, and intercepts?
• How do you sketch the curve `y=(x+1)/sqrt(5x^2+35)` by finding local maximum, minimum, inflection points, asymptotes, and intercepts?
• How do you sketch the curve `y=x^5-x` by finding local maximum, minimum, inflection points, asymptotes, and intercepts?
• How do you sketch the curve `y=x+1/x` by finding local maximum, minimum, inflection points, asymptotes, and intercepts?
• How do you sketch the curve `y=x^2+1/x` by finding local maximum, minimum, inflection points, asymptotes, and intercepts?
• How do you sketch the curve `y=(x+5)^(1/4)` by finding local maximum, minimum, inflection points, asymptotes, and intercepts?
• How do you sketch the curve `y=cos^2x-sin^2x` by finding local maximum, minimum, inflection points, asymptotes, and intercepts?
• How do you sketch the curve `y=x^3+6x^2+9x` by finding local maximum, minimum, inflection points, asymptotes, and intercepts?
• How do you sketch the curve `y=x/(x^2-9)` by finding local maximum, minimum, inflection points, asymptotes, and intercepts?
• How do you sketch the curve `y=x^2/(x^2+9)` by finding local maximum, minimum, inflection points, asymptotes, and intercepts?
• How do you sketch the curve `f(x)=1/(1+x^2)` by finding local maximum, minimum, inflection points, asymptotes, and intercepts?
• How do you sketch the curve `f(x)=1+1/x^2` by finding local maximum, minimum, inflection points, asymptotes, and intercepts?
• How do you sketch the graph `y=x^3-4x^2` using the first and second derivatives?
• How do you sketch the graph `y=x^4-27x` using the first and second derivatives?
• How do you sketch the graph `y=x^4+2x^2-3` using the first and second derivatives?
• How do you sketch the graph `y=x^5-16x` using the first and second derivatives?
• How do you sketch the graph `y=x^2/(1+x^2)` using the first and second derivatives?
• How do you sketch the graph `y=x+1/x` using the first and second derivatives?
• How do you sketch the graph `y=x^3+2x^2+x` using the first and second derivatives?
• How do you sketch the graph `y=x^4-x^3-x` using the first and second derivatives?
• How do you sketch the graph `y=x^4-2x^3+2x` using the first and second derivatives?
• How do you sketch the graph `y=sqrt(1+x^2)` using the first and second derivatives?
• How do you sketch the graph `y=1/(1+x^4)` using the first and second derivatives?
• How do you sketch the graph `y=sinx+sin^2x` using the first and second derivatives from `0<=x<2pi`?
• How do you sketch the graph `y=(2+sinx)^2` using the first and second derivatives from `0<=x<2pi`?
• How do you sketch the graph `y=e^(3x)-4e^x` using the first and second derivatives?
• How do you sketch the graph `y=(2e^x)/(1+e^(2x))` using the first and second derivatives?
• How do you sketch the graph `y=ln(1/x)` using the first and second derivatives?
• How do you find intercepts, extrema, points of inflections, asymptotes and graph `y=x/(x^2+1)`?
• How do you find intercepts, extrema, points of inflections, asymptotes and graph `y=(2x)/(x^2-1)`?
• How do you find intercepts, extrema, points of inflections, asymptotes and graph `y=(x+2)/x`?
• How do you find intercepts, extrema, points of inflections, asymptotes and graph `g(x)=x+32/x^2`?
• How do you find intercepts, extrema, points of inflections, asymptotes and graph `f(x)=(x^2+1)/x`?
• How do you find intercepts, extrema, points of inflections, asymptotes and graph `f(x)=(2x^2-5x+5)/(x-2)`?
• How do you find intercepts, extrema, points of inflections, asymptotes and graph `h(x)=xsqrt(9-x^2)`?
• How do you find intercepts, extrema, points of inflections, asymptotes and graph `y=xsqrt(16-x^2)`?
• How do you find intercepts, extrema, points of inflections, asymptotes and graph `y=2-x-x^3`?
• How do you find intercepts, extrema, points of inflections, asymptotes and graph `y=3x^4+4x^3`?
• How do you find intercepts, extrema, points of inflections, asymptotes and graph `y=3x^4-6x^2+5/3`?
• How do you find intercepts, extrema, points of inflections, asymptotes and graph `y=x^5-5x`?
• How do you find intercepts, extrema, points of inflections, asymptotes and graph `y=abs(2x-3)`?
• How do you find intercepts, extrema, points of inflections, asymptotes and graph `y=(20x)/(x^2+1)-1/x`?
• How do you find intercepts, extrema, points of inflections, asymptotes and graph `f(x)=x/sqrt(x^2+7)`?
• How do you find intercepts, extrema, points of inflections, asymptotes and graph `f(x)=(4x)/(sqrt(x^2+15))`?
• Consider the function `f(x)=e^(-x^(2))`?
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