Derivatives from Graphs and Tables
- Sketch the graph of f′ from the graph of f and read the correspondence in both directions
- Estimate a derivative from a table using a difference quotient
- Apply the rules of differentiation to values given only in a table
The correspondence to memorize
A large fraction of the exam presents a graph of f′ and asks about f, or the reverse, and the translation is mechanical once you hold the dictionary. Where f is increasing, f′ is positive — above the axis. Where f is decreasing, f′ is negative. Where f has a horizontal tangent (a local max, local min, or a flat spot), f′ is zero — crossing or touching the axis. Where f is concave up, f′ is increasing; where f is concave down, f′ is decreasing. Where f has an inflection point, f′ has a local extremum. Notice the shift in vocabulary: features of f described by sign become features of f′ described by position, and features of f described by concavity become features of f′ described by direction.
The trap that catches most students
When you are handed the graph of f′ and asked where f has a maximum, the answer is where f′ crosses from positive to negative — not where f′ itself has a maximum. A student reading the picture rather than the labels will point at the highest point on the displayed curve, which is where f is increasing fastest and is usually nowhere near a maximum of f. Before answering any such question, write at the top of your work which function is graphed. That one line prevents the single most common error in Units 2 and 5, and it costs nothing.
Tables: estimate with the closest available points
Given a table of f values, estimate f′(a) with a difference quotient, and prefer a symmetric one when the table allows it: f′(a) ≈ [f(a + h) − f(a − h)]/(2h), using the closest tabulated point on each side. If only one side is available — at the edge of the table — use a one-sided quotient and say that is what you did. Language matters on the free-response: because the table gives finitely many values, the honest verb is approximate or estimate, and a response that claims to have found f′(3) from a table has overstated what the data supports. When a question asks you to justify that some value of f′ is attained exactly, the tool is the MVT, not the table.
A table gives f(2) = 5, f′(2) = −3, g(2) = 4, g′(2) = 6. Find (a) the derivative of f·g at x = 2 and (b) the derivative of f/g at x = 2.
- 1.Product rule: (fg)′ = f′g + fg′. At x = 2 this is (−3)(4) + (5)(6).
- 2.(−3)(4) = −12 and (5)(6) = 30, so (fg)′(2) = −12 + 30 = 18.
- 3.Quotient rule: (f/g)′ = (f′g − fg′)/g². At x = 2 the numerator is (−3)(4) − (5)(6) = −12 − 30 = −42.
- 4.The denominator is g(2)² = 4² = 16.
- 5.So (f/g)′(2) = −42/16 = −21/8 = −2.625.
Table problems test whether you know the rules, not whether you can differentiate a formula — there is no formula. Write the general rule first, then substitute. Students who substitute first almost always drop a term of the product rule or reverse the subtraction in the quotient rule.
The graph of f′ is positive on (0, 3), zero at x = 3, and negative on (3, 6). What does f do at x = 3?
A table gives f(4) = 10 and f(4.2) = 10.9. The best available estimate of f′(4) from these two values is
Answer the 2 checkpoints as you read.
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