Grade by Grade
B. Derivatives and Their Applications · Curve Sketching: Connecting f, f′ and f″

Curve Sketching: Connecting f, f′ and f″: Unit Review

B1.1 B1.2 B1.3 B1.4 B1.5About 40 minutes
Learning goals
  • I can find and classify critical points, intervals of increase/decrease, concavity and inflection points.
  • I can connect the graphs of f, f′ and f″, and describe possible graphs from derivative information.
  • I can sketch polynomial and simple rational functions from their equations, including asymptotes.

Words to know

critical number
c in the domain with f′(c) = 0 or f′(c) undefined.
concavity
Whether the graph bends up (∪, f″ > 0) or down (∩, f″ < 0).
point of inflection
A point on the graph where the concavity changes.
asymptote
A line the graph approaches: vertical, horizontal or oblique.

Let's review!

In this unit you learned to read a graph through its derivatives. You sketched f′ from f, found critical numbers and intervals of increase and decrease, and classified extrema with the first derivative test. You met the second derivative as the rate of change of the rate of change, used it for concavity and points of inflection, and added the second derivative test. You worked backwards from derivative information to possible graphs, then sketched polynomials and, after studying asymptotes, rational functions from their equations.

The next unit, Applications of Derivatives, uses all of this for motion, related rates and optimization. This review has a few questions from every lesson. Try each one, then press Show answer to check.

How to use this review

Work through one or two sections at a time with paper and a pencil. For every function question, factor f′ and f″ before reading signs, put any x-values not in the domain into your charts, and always get y-values from the original f. For each question you miss, write down why (factoring? sign chart? substituted into the wrong function? forgot an asymptote?) and go back to that lesson. Finish with the 10-question quiz, then check one or two sketches with a graphing tool.

One function, three graphs, one sketchfvalues, intercepts,asymptotesf′rise / fall,max / minf″∪ / ∩,inflection pointssketchcombine alland verify
The unit in one line: f′ tells you where the graph rises and falls, f″ tells you how it bends, and together with intercepts and asymptotes they give a complete sketch.

Sketching the Graph of f′ from the Graph of f

1

A smooth f rises for x < 2 and falls for x > 2. Describe f′.

2

g(x) = f(x) + 7. Compare g′ and f′.

3

Where is the derivative of f(x) = |x + 3| undefined, and what is it elsewhere?

Intervals of Increase and Decrease, and Critical Numbers

1

Find the critical numbers of f(x) = x³ − 6x² + 5.

2

Find the intervals of increase and decrease of f(x) = x³ − 6x² + 5.

3

Where is f(x) = x + 16/x decreasing?

Local Maximums and Minimums: The First Derivative Test

1

Classify the critical points of f(x) = x⁴ − 2x².

2

f′(x) = (x − 3)²(x + 1). Classify x = −1 and x = 3.

3

Find the absolute extrema of f(x) = x³ − 3x on 0 ≤ x ≤ 2.

The Second Derivative: The Rate of Change of the Rate of Change

1

Depths 0, 3, 5, 6, 6.5 cm at t = 0 to 4 h. Concave up or down?

2

"Sales are falling, but more slowly." Signs of S′ and S″?

3

The graph of f′ has a local maximum at x = 2. What is at x = 2 on f?

Finding the Second Derivative of Polynomial and Rational Functions

1

Find f″ for f(x) = (4x − 3)³.

2

Find f″ for f(x) = 5/x².

3

Find f″ for f(x) = x/(x + 4).

Concavity and Points of Inflection

1

Find the inflection point of f(x) = x³ + 6x² − 1.

2

Find the intervals of concavity of f(x) = x⁴ − 6x³.

3

Does f(x) = x⁶ have an inflection point at 0?

The Second Derivative Test, and Connecting f, f′ and f″

1

Use the second derivative test on f(x) = x³ − 3x² − 24x.

2

f′(c) = 0 and f″(c) = 0. What should you do?

3

On an interval f′ is negative and increasing. Describe f.

Reading f from Information About f′ and f″

1

f′(x) = 4x³ and f(0) = −1. Find f.

2

Why do sign conditions on f′ and f″ allow infinitely many graphs?

3

The graph of f′ crosses the x-axis from + to − at x = 5 and has a minimum at x = 1. Describe f at x = 5 and x = 1.

Curve Sketching I: Polynomial Functions

1

Is f(x) = 2x⁵ − x³ even, odd or neither?

2

Sketch, in words, f(x) = x³ − 6x² + 9x.

3

An even quartic has a local min at (−2, −5). Name another key point.

Curve Sketching II: Quartics, Flat Points and Unfriendly Zeros

1

Find the key points of f(x) = x⁴ − 4x³ + 10.

2

How many zeros does f(x) = x⁴ − 4x³ + 10 have?

3

A cubic has max (0, −1) and min (2, −5). How many zeros?

Asymptotes of Rational Functions: Vertical, Horizontal and Oblique

1

Find the asymptotes of f(x) = (3x − 2)/(x + 5).

2

Find the oblique asymptote of f(x) = (2x² + x)/(x − 1).

3

Find the hole of f(x) = (x² − 25)/(x + 5).

Curve Sketching III: Rational Functions

1

For f(x) = x²/(x − 3), find the asymptotes.

2

For f(x) = x²/(x − 3), find the extrema.

3

For f(x) = 3/(x² + 3), find the extremum and the HA.

f(x) = x³ − 3xxy−2−112−4−3−2−1123maxminIPf′(x) = 3x² − 3xy−2−112−426f″(x) = 6xxy−2−112−8−426
A quick self-test: cover the labels and explain why each dot on f′ and f″ lines up with a feature of f.
Just for fun

The unit in one sentence: f went for a walk, f′ described every hill, and f″ complained about every bend.

Check your understanding

Quiz

1.

Where is f(x) = x³ − 12x decreasing?

2.

f′ changes from + to − at x = 1. At x = 1, f has:

3.

What is f″(x) for f(x) = 2x⁴ − x²?

4.

Where is f(x) = x³ − 9x² concave up?

5.

f′(2) = 0 and f″(2) = −4. What is at x = 2?

6.

The graph of f′ has a local minimum at x = −3. On the graph of f, x = −3 is:

7.

f′(x) = 3x² − 2 and f(0) = 4. What is f(x)?

8.

Which function is odd?

9.

What is the oblique asymptote of f(x) = (x² + 2x + 3)/(x + 1)?

10.

Why is there no point of inflection at x = 0 for f(x) = 1/x even though its concavity changes?

For parents and teachers

How to use the results: look at which kinds of questions your teen missed and send them back to the matching lesson.

  • Mixing up heights and slopes when sketching f′ → Sketching the Graph of f′ from the Graph of f.
  • Factoring f′, sign charts, or joining intervals across an asymptote → Intervals of Increase and Decrease, and Critical Numbers.
  • Calling every critical number a max or min, or forgetting endpoints → Local Maximums and Minimums: The First Derivative Test.
  • Confusing concave up with increasing, or "falling faster" with f″ > 0 → The Second Derivative: The Rate of Change of the Rate of Change.
  • Algebra errors in f″, especially with negative exponents or the quotient rule → Finding the Second Derivative of Polynomial and Rational Functions.
  • Declaring an inflection point wherever f″ = 0 → Concavity and Points of Inflection.
  • Misusing the second derivative test when f″(c) = 0 → The Second Derivative Test, and Connecting f, f′ and f″.
  • Forgetting + C or treating the graph of f′ as f → Reading f from Information About f′ and f″.
  • Incomplete sketches (missing symmetry, intercepts or y-values) → Curve Sketching I and Curve Sketching II.
  • Asymptote errors (holes vs vertical asymptotes, degree rules, oblique asymptotes) → Asymptotes of Rational Functions.
  • Rational-function sketches → Curve Sketching III: Rational Functions.

The next unit, Applications of Derivatives, assumes all of these skills, so it is worth fixing gaps now.