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Year 12>
Derivative Curve Sketching>
Increasing and Decreasing Intervals>
Increasing and Decreasing IntervalsIncreasing and Decreasing Intervals
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Question 1 of 4
1. Question
Create a sign diagram for the curve `f(x)=x^2`
Hint
Help VideoCorrect
Excellent
Incorrect
A sign diagram is a way of visualizing a curve’s graph by indicating the inflection points, stationary points, and the increases or decreases in the curve.First, set up a horizontal line with a matching indicator of either a stationary or an inflection point from the graph
Notice that a stationary/turning point exists at `0`. Hence, the line is marked `0` at a location that matches the graphNext, get the derivative of the function to identify the gradients of parts of the curve`f(x)` `=` `x^2` `f'(x)` `=` `2x` Note that if `x≥0` or positive, `f'(x)` is also positive, which means the curve’s slope is increasingIndicate this on the sign diagram by adding a positive sign to the right of `0`
Also note that if `x≤0` or negative, `f'(x)` is also negative, which means the curve’s slope is decreasingIndicate this on the sign diagram by adding a negative sign to the left of `0`
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Question 2 of 4
2. Question
Create a sign diagram for the curve `f(x)=x^3`
Hint
Help VideoCorrect
Well Done
Incorrect
A sign diagram is a way of visualizing a curve’s graph by indicating the inflection points, stationary points, and the increases or decreases in the curve.First, set up a horizontal line with a matching indicator of either a stationary or an inflection point from the graph
Notice that a stationary/turning point exists at `0`. Hence, the line is marked `0` at a location that matches the graphNext, get the derivative of the function to identify the gradients of parts of the curve`f(x)` `=` `x^3` `f'(x)` `=` `3x^2` Note that since the power of `x` is even, `f'(x)` will always be positive regardless of the sign of `x`. This means the curve’s slope is always increasingIndicate this on the sign diagram by adding a positive sign to the left and right of `0`
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Question 3 of 4
3. Question
Create a sign diagram for the curve below
Hint
Help VideoCorrect
Fantastic
Incorrect
A sign diagram is a way of visualizing a curve’s graph by indicating the inflection points, stationary points, and the increases or decreases in the curve.First, set up a horizontal line with a matching indicator of either a stationary or an inflection point from the graph
Notice that an inflection point exists at `0` and a stationary point exists at `3`. Hence, the line is marked `0` and `3` at a location that matches the graphNext, notice that the curve’s slope decreases at the left side of `0`Indicate this on the sign diagram by adding a negative sign to the left of `0`
Also, the curve’s slope decreases at the right side of `0`Indicate this on the sign diagram by adding a negative sign to the right of `0`
Lastly, notice that the curve’s slope increases at the right side of `3`Indicate this on the sign diagram by adding a positive sign to the right of `3`
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Question 4 of 4
4. Question
Create a sign diagram for the curve below
Hint
Help VideoCorrect
Great Work
Incorrect
A sign diagram is a way of visualizing a curve’s graph by indicating the inflection points, stationary points, and the increases or decreases in the curve.First, set up a horizontal line with a matching indicator of either a stationary or an inflection point from the graph
Notice that an inflection point exists at `0` and a stationary point exists at `-1` and `2`. Hence, the line is marked `-1,0` and `2` at a location that matches the graphNext, notice that the curve’s slope increases at the left side of `-1`Indicate this on the sign diagram by adding a positive sign to the left of `-1`
Then, the curve’s slope decreases at the right side of `-1`Indicate this on the sign diagram by adding a negative sign to the right of `-1`
Still, the curve’s slope decreases at the right side of `0`Indicate this on the sign diagram by adding a negative sign to the right of `0`
Lastly, notice that the curve’s slope increases at the right side of `2`Indicate this on the sign diagram by adding a positive sign to the right of `2`
