Curve Sketching Calculator

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Example

Created on 2024-06-20Asked by Penelope Jackson (Solvelet student)

Sketch the curve defined by the function

f(x) = x^3 - 3x^2 - 4x + 12

Solution

Step 1: Identify the function

f(x) = x^3 - 3x^2 - 4x + 12

. Step 2: Find the critical points by setting the derivative equal to zero and solving for

x

f'(x) = 3x^2 - 6x - 4 = 0.

Step 3: Solve the quadratic equation:

x^2 - 2x - \frac{4}{3} = 0.

Step 4: Find the roots using the quadratic formula or factoring:

x = \frac{2 \pm \sqrt{4 + 4}}{2} = \frac{2 \pm \sqrt{8}}{2} = 1 \pm \sqrt{2}.

Step 5: Determine the concavity by analyzing the second derivative:

f''(x) = 6x - 6.

Step 6: Analyze the behavior of

f(x)

around the critical points and inflection points. Step 7: Sketch the curve using the information obtained from the steps above. Step 8: Conclusion. The sketch of the curve

f(x) = x^3 - 3x^2 - 4x + 12

is completed. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Mason Lewis on Solvelet

1. Construct the graph of

f(x) = x^3 - 3x^2 - 9x + 5

, indicating intercepts, asymptotes, extrema, and concavity 2. Analyze the behavior of the function

g(x) = \frac{x^2 - 4}{x - 2}

and sketch its graph, identifying intercepts, asymptotes, extrema, and concavity.,

DefinitionCurve sketching is a method in calculus and analytical geometry that entails examining the behavior of a function to draw its graph in a precise manner. It makes you work out which correspond to the domain, range, intercepts, symmetries and asymptotes, critical points of the function (not what the function is, mind you, just information about that function) and sketch it simply from that. Use curve sketching to graphically display when and where (on the graph of the function) functions behave in a particular manner, like having maxima and minima, or inflection points. It is a graph representation of functions for the purpose of analysis and interpretation. For instance, given a rational function, it is typical to search for vertical and horizontal asymptotes, intercepts, and intervals on which it is increasing or decreasing, and graph and connect the dots thus found by hand.

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