The definition of the derivative practice question 9
The function is piecewise linear: its graph is made of straight segments joining , , and , in that order, as shown. Which statement about its derivative is correct?
On a straight segment the slope is the same at every point. Which segment does each of x = 0, 2 and 4 lie on?
Slope is rise over run between the two ends of a segment. At x = 1 two segments meet: do they have the same slope there?
Theory UBC Math 100 §3.1Key results
Worked solution
Strategy
Each segment is a line, so f′ is that line’s slope at every point inside it: rise over run between its ends. Where two segments with different slopes meet, the graph has a corner and f′ does not exist.
Working
x = 0 lies on the first segment, so f′(0) = 2.
The level segment has slope 0, so f′(2) = 0.
The falling segment has a negative slope, so f′(4) = −4.
At x = 1 the slope is 2 on the left and 0 on the right. The difference quotient has different limits from the two sides, so f′(1) does not exist, even though f is continuous there.
Answer
The answer is f′(0) = 2, f′(2) = 0, f′(4) = −4, and f′(1) does not exist because the graph has a corner there..
Check
The slopes rebuild the graph: from height 1, climbing 2 per unit for 3 units reaches 7, level for 2 units stays at 7, and falling 4 per unit for 2 units ends at −1.
Where this goes wrong
Giving f′(1) a value because the graph does not break at x = 1. A corner is where two different slopes meet; there is no one tangent line there, so the derivative does not exist.
The idea behind it
The definition of the derivative lesson, a figure to move, in a few steps
Leans on
Slope of a line through two points, from Functions, graphs and inverses
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