How do you sketch the graph #y=x^4-x^3-x# using the first and second derivatives?
See below
Using the power rule
This is a polynomial of degree 3. To find zeros for polynomials of degree 3 or higher we use Rational Root Test.
To find any other real roots:
To find other zeros:
graph{x^4-x^3-x [-10, 10, -5, 5]}
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To sketch the graph of (y = x^4 - x^3 - x) using the first and second derivatives, follow these steps:
- Find the critical points by setting the first derivative equal to zero and solving for (x).
- Use the second derivative test to determine the nature of each critical point (whether it's a local minimum, maximum, or inflection point).
- Plot the critical points on the graph.
- Determine the behavior of the function as (x) approaches positive and negative infinity.
- Sketch the graph, incorporating the information from steps 1-4.
Let's go through each step:
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Find the first derivative of (y = x^4 - x^3 - x) and set it equal to zero: [ \frac{dy}{dx} = 4x^3 - 3x^2 - 1 = 0 ] Solve for (x).
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Once you find the critical points, (x_1), (x_2), and (x_3), evaluate the second derivative at these points: [ \frac{d^2y}{dx^2} = 12x^2 - 6x ] Evaluate ( \frac{d^2y}{dx^2} ) at each critical point.
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Use the second derivative test:
- If ( \frac{d^2y}{dx^2} > 0 ), the critical point is a local minimum.
- If ( \frac{d^2y}{dx^2} < 0 ), the critical point is a local maximum.
- If ( \frac{d^2y}{dx^2} = 0 ), the test is inconclusive.
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Determine the behavior of the function as (x) approaches positive and negative infinity by observing the leading term of the function (x^4).
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Sketch the graph based on the information obtained from steps 1-4, including the critical points and the behavior of the function at positive and negative infinity. Make sure to accurately represent the concavity of the function based on the signs of the second derivative.
Once you've followed these steps, you should have a rough sketch of the graph of (y = x^4 - x^3 - x).
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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- What is the second derivative of #y=x*sqrt(16-x^2)#?
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