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Calculus IApplications

Read the derivatives, see the shape.

Use slopes to find where a curve rises, falls, turns and bends. Then practise every question from the notes, including reading a function from its derivative graph.

Rising, falling and turning · Step 1 of 6

A derivative tells you which way the graph is moving

−2−1012−3−2−10123
Read left to right. A positive slope means the height rises; a negative slope means it falls. A critical number is an input in the domain where the derivative is zero or does not exist. Split the domain at those inputs and at any gaps, then test one point in each interval.
Try it: Move along f(x)=x3−3x{f(x)} = {x^3-3x}. Check the tangent direction against the highlighted interval. Zero slope by itself does not tell you what happens on either side.
Input x0.0
(−∞,−1){(-\infty,-1)}
f′:+{f':\quad +}
(−1,1){(-1,1)}
f′:−{f':\quad -}
(1,∞){(1,\infty)}
f′:+{f':\quad +}
f′(x)=3x2−3=3(x−1)(x+1){\displaystyle f'(x)} = {\displaystyle 3x^2} - {\displaystyle 3} = {\displaystyle 3(x-1)(x+1)}

Test −2, 0 and 2: the derivative is 9, −3 and 9. Use open intervals for this sign chart. Never join intervals across a gap in the domain.

f′(0.0)=3(0.0)2−3=−3.00{\displaystyle f'(0.0)} = {\displaystyle 3(0.0)^2} - {\displaystyle 3} = {\displaystyle -3.00}
Check yourself
If the derivative is negative throughout an interval, what does the function do?

Need help turning derivatives into a graph?

Bring your practice questions. We work through them together until every type feels routine.

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