How do you use the first and second derivatives to sketch #h(x)=x³-3x+1#?
First derivative will give us critical points:
Points that represent potential local maximum/minimum values or aymptotes
Plugging in values on either side of these critical points will show how the curve is acting (ie. positive/negative slopes)
Second derivative will give us inflection points:
Points in which the curve changes concavity
...As well as tell us if the curve is concave up/down.
So, lets first take the derivative of Check the intervals around the critical values: For For For This should make sense since we started with a cubic function. These changes in slope indicate local extrema. Local maximum at Now, lets take the second derivative: Check the intervals around this value. For For Thus, there is an inflection point at x=0. Now that we have all that information, we should gather up some basic information to help us plot. Let's find some intercepts by plugging in 0 for x into the original function f(x). y-intercept Plug in our extrema values and plot those.
Now, we know how the graph acts based on those intervals: Increasing from
Decreasing from
Now concavity; since we know that the function is differentiable along the Concave down from graph{x^3-3x+1 [-5.304, 5.796, -2, 3.546]}
Local minimum at
Concave up from
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To sketch the function ( h(x) = x^3 - 3x + 1 ) using its first and second derivatives:
- Find the first derivative ( h'(x) ) by differentiating ( h(x) ) with respect to ( x ).
- Find the critical points by setting ( h'(x) ) equal to zero and solving for ( x ). These points represent potential local extrema or inflection points.
- Determine the intervals of increase and decrease of ( h(x) ) by examining the sign of ( h'(x) ) in between the critical points.
- Find the second derivative ( h''(x) ) by differentiating ( h'(x) ) with respect to ( x ).
- Determine the concavity of ( h(x) ) by examining the sign of ( h''(x) ) between critical points.
- Locate any points of inflection where ( h''(x) = 0 ) or is undefined.
- Sketch the graph, incorporating the information about critical points, intervals of increase/decrease, and concavity.
By following these steps, you can construct a rough sketch of the function ( h(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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- How do you use the first and second derivatives to sketch #y=(x^3)-(6x^2)+5x+12#?
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