Sketching accumulation functions connects a rate graph to its total change. By interpreting features of f(t), students visualize how the integral builds the graph of g(x).

Using Graphs to Sketch Accumulation Functions

When an accumulation function is defined as g(x) = ∫ₐˣ f(t) dt, its graph reflects how the area under f develops as x moves along the horizontal axis. Understanding how to translate the shape and behavior of f into corresponding characteristics of g is central to this subsubtopic.

The Structure of an Accumulation Function

An accumulation function begins at a fixed lower limit a, meaning that g(a) always equals zero because no area has accumulated yet. As the upper limit x increases, g(x) grows or decreases depending on whether the graph of f(t) lies above or below the x-axis. This relationship highlights how the independent variable in the integral controls the accumulation process.

Because g(x) measures net area, regions where f is positive increase g, and regions where f is negative decrease g.

Diagram showing positive and negative signed areas under a function $f(x)$, illustrating how net area determines the change in an accumulation function such as $g(x) = \int_a^x f(t),dt$. The plus and minus symbols simply reinforce region signs and add no extra syllabus content. Source.

This perspective provides a powerful connection between geometric area and functional behavior.

Using Key Features of f to Understand g

The graph of f contains all the information needed to sketch g, and students must learn how to interpret that information systematically. Several foundational relationships guide this process:

• Where f(t) > 0, g(x) is increasing, because positive area is being added.

• Where f(t) < 0, g(x) is decreasing, because negative area decreases the accumulated total.

• Where f(t) = 0, g′(x) = 0, meaning g has a horizontal tangent at that point.

These relationships come directly from the Fundamental Theorem of Calculus, which states that an accumulation function’s derivative equals the original rate function.

Connecting g’s Shape to f’s Behavior

Students often benefit from analyzing qualitative features of f to forecast the overall shape of g. Important connections include:

• Maxima of g occur where f changes from positive to negative, because the accumulated area peaks before it starts decreasing.

• Minima of g occur where f changes from negative to positive, signaling a turnaround from decreasing to increasing accumulated area.

• Inflection points of g correspond to local extrema of f, because concavity in g is tied to whether f is increasing or decreasing.

These relationships stem from the fact that g′ = f and g″ = f′, making the analysis of g directly dependent on observable features of f.

Zeros and Net Accumulation

Although g(a) = 0 by definition, the accumulation function may cross the x-axis again if the total accumulated area returns to zero after being positive or negative. This occurs when the net signed area from a to x equals zero. Understanding how positive and negative regions of f balance helps students accurately track when g(x) equals zero again.

Interpreting and Sketching g from the Graph of f

When constructing g from a graph of f, a structured approach improves accuracy:

• Identify the starting point: Mark g(a) = 0.

• Analyze the sign of f: Determine intervals where f is above or below the x-axis to establish where g rises or falls.

• Note changes in sign: These indicate possible maxima or minima of g, depending on the transition direction.

• Assess the magnitude of area: Larger magnitudes of f(t) cause steeper increases or decreases in g.

• Track concavity: Where f is increasing, g is concave up; where f is decreasing, g is concave down.

• Watch for zero crossings of g: These occur where accumulated positive and negative areas cancel.

This systematic interpretation transforms geometric observations into an accurate qualitative sketch of the accumulation function.

A graph of $f(t)$ with shaded intervals and a corresponding table of accumulation values, illustrating how net area across successive intervals determines the values of $g(x)=\int_a^x f(t),dt$. The table includes numerical values not required by the syllabus but demonstrates the accumulation process described in the notes. Source.

Rate Magnitude and Slope of g

Because the slope of g at any point equals f(x), the steepness of g’s graph is directly tied to the height of the graph of f. Important implications include:

• Larger positive values of f create steeper upward slopes of g.

• Larger negative values of f create steeper downward slopes.

• When f is zero, the slope of g is zero, and g momentarily levels off.

Understanding this connection helps students interpret how rapidly accumulated change is occurring.

Visualizing Accumulation Across Intervals

Accumulation is best understood as a continuous process. Even when the graph of f is irregular or consists of disconnected segments, the accumulation function smooths these features into a continuously evolving curve. Students must think about net area at every step: even small positive regions contribute to a gentle upward bend in g, while brief negative regions introduce subtle decreases.

This perspective reinforces that sketching an accumulation function is not merely about plotting points but about developing an intuitive understanding of how rate information shapes total change.