Reading Derivatives from Graphs and Data: Sign, Magnitude, and Shapes

1) The derivative as a “slope sensor” on a graph

When you look at a graph of y=f(x), the derivative f'(x) tells you what the graph is doing right there: whether it is rising or falling, and how steeply. This chapter focuses on reading that information directly from shapes, and on estimating it from data tables.

Sign of f'(x): increasing vs. decreasing

• f'(x) > 0: the graph of f is increasing (rising as you move right).
• f'(x) < 0: the graph of f is decreasing (falling as you move right).
• f'(x) = 0: the tangent is horizontal. This often happens at local maxima/minima, but can also happen at “flat” points where the function keeps increasing or decreasing.

Geometrically, the sign is about the direction of the tangent line. A tangent that tilts upward to the right corresponds to positive slope; downward corresponds to negative slope.

Magnitude |f'(x)|: steepness

• Large |f'(x)|: very steep graph (rapid change in f per unit of x).
• Small |f'(x)|: nearly flat graph (slow change).
• |f'(x)| near 0: almost horizontal tangent.

Two points can both have positive derivative, but one can be much steeper: for example, a tangent slope of +5 is steeper than +0.5.

2) Where derivatives fail: corners, cusps, and vertical tangents

Some shapes prevent a single well-defined tangent slope at a point. On a graph, these are often visible as “sharp” or “too steep” features.

Corners (sharp turns)

At a corner, the left-hand slope and right-hand slope are different. The graph has a pointy change in direction, so there is no single tangent line that matches both sides.

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• Graph clue: looks like two line segments meeting with different slopes.
• Derivative clue: f'(x) does not exist at the corner.

Cusps (pointed tips)

A cusp is sharper than a corner: slopes may blow up in magnitude with opposite signs as you approach the point.

• Graph clue: a pointed “needle” tip.
• Derivative clue: f'(x) does not exist at the cusp.

Vertical tangents

Sometimes the curve becomes so steep that the tangent line is vertical. A vertical line has undefined slope, so the derivative is not a finite number there.

• Graph clue: the curve looks nearly vertical at a point.
• Derivative clue: f'(x) may be undefined or can be thought of as “infinite slope” (not a real-number derivative).

Quick classification task

For each feature below, decide whether f'(a) exists as a real number.

• A smooth rounded peak (like the top of a hill).
• A V-shaped bottom.
• A smooth curve that becomes vertical at one point.

Write one sentence justifying each answer using “single tangent slope” language.

3) Connecting f and f': key features to match

A powerful skill is translating between the graph of a function and the graph of its derivative. You do not need exact formulas; you focus on structure.

A checklist for sketching f'(x) from a graph of f(x)

• Step 1: Mark where f has horizontal tangents. These are candidates for f'(x)=0 (zeros of the derivative).
• Step 2: Determine where f is increasing/decreasing. Increasing intervals become f'(x)>0; decreasing intervals become f'(x)<0.
• Step 3: Compare steepness. Where f is steep, |f'| should be large; where f is flat, |f'| should be small.
• Step 4: Identify nondifferentiable points. Corners/cusps/vertical tangents in f become points where f' is undefined (often shown as breaks or open circles in a sketch).
• Step 5: Track how slopes change. If the slope of f is increasing as you move right, then f' is increasing; if slopes are decreasing, then f' is decreasing.

Interpreting “slopes increasing or decreasing” (shape of f')

Think of walking along the curve from left to right and watching the tangent slope value.

• If tangent slopes go from -3 to -1 to 0 to 2, then f'(x) is increasing on that interval.
• If tangent slopes go from 4 to 2 to 1 to 0, then f'(x) is decreasing on that interval.

This is a “graph-of-the-slope” viewpoint: f'(x) records the slope values you would write down at each x.

4) Guided sketching problems: from f to f'

Problem A: Smooth hill then valley

Suppose f(x) is smooth and has: a local maximum at x=-2, then decreases until a local minimum at x=1, then increases after that. The graph is steepest around x=0 and flatter near the max/min.

Task: Sketch a plausible f'(x).

• Step 1 (zeros): Put f'(x)=0 at x=-2 and x=1.
• Step 2 (sign): Since f increases before -2, set f'(x)>0 for x<-2. Since f decreases between -2 and 1, set f'(x)<0 on (-2,1). Since f increases after 1, set f'(x)>0 for x>1.
• Step 3 (magnitude): Make f'(x) most negative near x=0 (steepest downward slope), and closer to 0 near x=-2 and x=1 (flatter).
• Step 4 (shape): Draw a curve for f' that crosses the x-axis at -2 and 1, dips to a minimum near 0, and rises afterward.

Short explanation to write: “f' is positive where f rises, negative where f falls, and zero at horizontal tangents; its magnitude reflects steepness.”

Problem B: Corner point

Suppose f(x) is decreasing linearly for x<0 with slope -2, then increasing linearly for x>0 with slope +3, meeting at a sharp corner at x=0.

Task: Sketch f'(x) and describe what happens at x=0.

• For x<0, f'(x)=-2 (a horizontal line at -2).
• For x>0, f'(x)=3 (a horizontal line at 3).
• At x=0, f'(0) does not exist (jump in slope from -2 to 3).

Short explanation to write: “A corner has two different one-sided slopes, so there is no single derivative value at the corner.”

Problem C: Vertical tangent

Suppose f(x) is smooth except at x=2, where the curve becomes vertical (the graph looks like it shoots straight up as it passes through x=2).

Task: Describe what you expect for f'(x) near x=2.

• As x approaches 2, the slopes become extremely large in magnitude.
• The derivative at x=2 is not a finite real number, so f'(2) is undefined in the usual sense.
• A sketch of f' would show values growing without bound (or a break) near x=2.

5) Going the other way: sketching f from a graph of f'

If you are given f'(x), you can reconstruct the shape of f (up to a vertical shift) by interpreting derivative information.

A checklist for sketching f from f'

• Step 1: Find where f'(x)=0. These are points where f has horizontal tangents.
• Step 2: Use the sign of f'. If f'>0, f increases; if f'<0, f decreases.
• Step 3: Use the size of f'. Large positive f' means f rises steeply; small f' near 0 means f is nearly flat.
• Step 4: Look for changes in f'. If f' is increasing, the slopes of f are getting larger (the graph of f is “bending upward” in slope). If f' is decreasing, slopes are getting smaller.

Problem D: Derivative sign chart to function shape

You are told that:

• f'(x)>0 on (-∞,-1)
• f'(x)=0 at x=-1
• f'(x)<0 on (-1,2)
• f'(x)=0 at x=2
• f'(x)>0 on (2,∞)

Task: Sketch a plausible f(x) and label what happens at x=-1 and x=2.

• Since f' changes from positive to negative at -1, f has a local maximum at x=-1.
• Since f' changes from negative to positive at 2, f has a local minimum at x=2.
• Between them, f decreases; outside them, f increases.

Short explanation to write: “Where f' is positive the function rises; where negative it falls; sign changes at zeros indicate turning points.”

6) Estimating derivatives from tables (numerical slopes)

When you have data rather than a formula, you estimate f'(x) using average rates of change over small intervals. The basic tool is the difference quotient computed from nearby table values.

Average rate over a small interval

Given values (x, f(x)) and (x+h, f(x+h)), the average rate of change is:

average rate from x to x+h = [f(x+h) - f(x)] / h

If h is small and the data are smooth, this can approximate f'(x).

Forward, backward, and central estimates

• Forward difference: [f(x+h)-f(x)]/h
• Backward difference: [f(x)-f(x-h)]/h
• Central difference: [f(x+h)-f(x-h)]/(2h) (often more accurate when data are smooth and evenly spaced)

Worked table example (step-by-step)

Suppose you have the table:

Estimate f'(2.0) using different methods.

• Forward difference (h=0.1): [f(2.1)-f(2.0)]/0.1 = (4.41-4.00)/0.1 = 0.41/0.1 = 4.1
• Backward difference (h=0.1): [f(2.0)-f(1.9)]/0.1 = (4.00-3.61)/0.1 = 0.39/0.1 = 3.9
• Central difference (h=0.1): [f(2.1)-f(1.9)]/0.2 = (4.41-3.61)/0.2 = 0.80/0.2 = 4.0

Interpretation: the slope near x=2 is about 4. The forward and backward estimates differ slightly because they sample different sides of the point.

Sensitivity to step size h

Choosing h is a tradeoff:

• If h is too large, the average rate may miss local behavior (it “smears” the slope over a wide interval).
• If h is very small, measurement noise or rounding can dominate the subtraction f(x+h)-f(x), making the estimate unstable.

Mini-task: If a sensor reports f(x) rounded to the nearest 0.01, explain in 1–2 sentences why using h=0.001 could produce unreliable derivative estimates.

Noisy data: what to do

With noise, derivative estimates can fluctuate wildly because differentiation amplifies small wiggles in the data.

• Use central differences when possible (they average information from both sides).
• Use a slightly larger h to reduce noise sensitivity, while staying local enough to reflect the shape.
• Look for trends (sign and relative magnitude) rather than expecting a perfectly smooth derivative curve.

7) Guided interpretation problems (short written explanations)

Problem E: Reading sign and zeros from a graph description

A graph of f is smooth. It rises from x=-4 to x=-1, is flat at x=-1, then falls until x=3, is flat again at x=3, then rises after 3.

• Question 1: On which intervals is f'(x) positive? negative?
• Question 2: Where does f'(x)=0?
• Write-up (2–3 sentences): Explain how you know using increasing/decreasing language.

Problem F: Matching steepness

On (0,1) the graph of f rises gently; on (1,2) it rises very steeply; on (2,3) it still rises but begins to flatten.

• Task: Describe how f'(x) compares across these intervals (small positive vs. large positive, increasing vs. decreasing).
• Write-up (1–2 sentences): Use the phrase “magnitude of the derivative” correctly.

Problem G: Detecting nondifferentiability from a sketch

You see a curve that is smooth everywhere except at x=5, where it has a sharp corner.

• Question: What happens to f'(5)?
• Write-up (1–2 sentences): Mention one-sided slopes and why they prevent a single tangent slope.

Problem H: Estimating f'(x) from a data table and commenting on reliability

Given:

• Task 1: Estimate f'(11) using a central difference with h=1.
• Task 2: Estimate f'(11) using a forward difference with h=1.
• Task 3: In 2–3 sentences, discuss why the two estimates differ and what that suggests about the local behavior or noise.