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AI HL · 12.9

Differentiation

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12.9.1

Gradient, increasing and decreasing

12.9.1

Gradient, increasing and decreasing

Target: I can say that the gradient of a curve at a point is the gradient of the tangent at that point, and estimate it from a graph. (+3 more)

By the end of this section

Key points

The gradient of a curve

  • The gradient of a curve at a point is the gradient of the tangent to the curve at that point.
  • Along a curve the gradient changes. Where the curve is steep the gradient is large; where it flattens the gradient is small.
  • At a turning point the tangent is horizontal, so the gradient is zero.

Increasing and decreasing

  • f is increasing where the graph rises as you move right: the gradient is positive.
  • f is decreasing where the graph falls as you move right: the gradient is negative.
  • A function does not have to do one thing everywhere. Find the intervals, then write each one as an inequality in x.
  • The turning points cut the x-axis into intervals. Check one interval at a time.

Reading intervals from a graph

  • Intervals are in x. Do not give the y-coordinates of the turning points.
  • Use \(<\) for the ends that are turning points. If the question gives a domain such as \(0\le x\le 360\), keep its ends with \(\le\).
  • For a trig graph, find the maxima and minima first. Between them the graph is either rising or falling.

Explore: the tangent at a point

Drag the orange point P along the curve. The orange line is the tangent at P. Its gradient is f′(x) at that point.


Curve
Point
Gradient at P
Tangent (orange)

The two axes have different scales, so the normal does not always look perpendicular on the screen.

Drag P along the curve.

  1. On the quadratic, move P to the lowest point. What is the gradient there? What does the tangent look like?
  2. On the cubic, where is the gradient negative? What is the curve doing there? Why?
  3. Show the normal. Multiply the gradient of the tangent by the gradient of the normal. What do you get? Why does a horizontal tangent break the rule?
  4. On the curve y = 4x + 25/x, what happens to the gradient as x gets larger? Why?

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
A cubic with two turning points

The graph of y = f(x) is a cubic with a local maximum at x = −1 and a local minimum at x = 2. Write down the values of x for which f is decreasing.

−3−2−11234−10−55xyy = f(x)(−1, 3.5)(2, −10)
Working
  1. Turning points

    Gradient is zero at \(x=-1\) and \(x=2\)

    A turning point has a horizontal tangent. These two x-values cut the graph into three pieces. Use the x-values, not the heights.

  2. Left of the maximum

    For \(x<-1\) the graph rises: f is increasing

    Gradient positive: the curve goes up as you move right.

  3. Between the turning points

    For \(-1<x<2\) the graph falls: f is decreasing

    Gradient negative: the curve goes down as you move right.

  4. Right of the minimum

    For \(x>2\) the graph rises again: f is increasing

    A function does not have to do one thing everywhere.

  5. State

    Decreasing: \(-1<x<2\); increasing: \(x<-1\) or \(x>2\)

    Write each interval as an inequality in x.

Video walkthrough coming
A parabola's vertex

The graph shows y = f(x), a parabola with its lowest point at (3, −2). Write the values of x for which f is increasing.

−11234567−2246xyy = f(x)(3, −2)
Working
  1. The turning point

    Vertex at \(x=3\)

    The gradient is zero here, and it is where the graph changes direction.

  2. Left of the vertex

    For \(x<3\) the graph falls: decreasing

    Reading left to right, the curve goes down.

  3. Right of the vertex

    For \(x>3\) the graph rises: increasing

    Reading left to right, the curve goes up.

  4. State

    f is increasing for \(x>3\)

    The inequality uses x. The −2 only tells you the height of the vertex.

Video walkthrough coming
A wave from 0 to 360

The graph shows y = f(x) for 0 ≤ x ≤ 360, where x is in degrees. It has a minimum at (90, 2) and a maximum at (270, 6). Write the values of x for which f is increasing.

901802703602468x (degrees)yy = f(x)(90, 2)(270, 6)
Working
  1. Turning points

    Gradient is zero at \(x=90\) and \(x=270\)

    The graph starts at 4, so it is not at a turning point at x = 0.

  2. Start of the graph

    For \(0\le x<90\) the graph falls

    It goes down from 4 to the minimum.

  3. Middle

    For \(90<x<270\) the graph rises

    From the minimum up to the maximum.

  4. End of the graph

    For \(270<x\le 360\) the graph falls

    Keep the ends of the domain in your inequalities.

  5. State

    Increasing: \(90<x<270\)

    One interval only.

Video walkthrough coming
Estimating a gradient

The line shown is the tangent to the curve at P(4, 4). It passes through (2, 0). Find the gradient of the curve at P.

−2−1123456−22468xyy = f(x)P
Working
  1. Use the tangent

    The gradient of the curve at P is the gradient of the tangent at P

    The tangent touches the curve at P and has the same steepness there.

  2. Two points on it

    (2, 0) and (4, 4)

    Read two points on the line, not on the curve.

  3. Rise over run

    \(\dfrac{4-0}{4-2}=2\)

    Positive: the curve is rising at P.

  4. Where is the gradient zero?

    At the lowest point, \(x=0\)

    A horizontal tangent has gradient 0. You can name this without measuring anything.

Video walkthrough coming