# Curve Sketching

Mathematics I · Differential Calculus · https://tryals.app/learn/mathematics-i/curve-sketching

## The Sketching Toolkit

**Curve sketching** translates an abstract formula into an accurate graph by extracting features systematically. The first derivative builds the up-and-down skeleton, while the second adds bending.

| Feature | What to Look For | Formula / Rule |
| --- | --- | --- |
| **Intercepts** | Axes crossings | $x$ where $f(x)=0$; $y$ at $f(0)$ |
| **Asymptotes** | Boundary lines | Vertical where $f \to \pm\infty$; horizontal as $x \to \pm\infty$ |
| **Symmetry** | Axis mirroring | Even: $f(-x)=f(x)$; Odd: $f(-x)=-f(x)$ |

Working through this checklist ensures no critical behavior is missed before drawing lines.

## Derivatives and Shape

Derivatives reveal how a curve rises, falls, and bends between its structural boundaries.

| Derivative | Sign | Graphical Meaning |
| --- | --- | --- |
| **First ($f'$)** | Positive | Function is rising (increasing) |
| **First ($f'$)** | Negative | Function is falling (decreasing) |
| **Second ($f''$)** | Positive | **Concave up** (cups upward) |
| **Second ($f''$)** | Negative | **Concave down** (cups downward) |

**Inflection points** occur where the second derivative changes sign, marking where concavity shifts.

> **Pitfall:** Relying on $f'$ alone gets turning points right but misses **concavity** and **inflection points**, ruining the true shape. Both derivatives are required.

## Practice questions

6 of this lesson's 11 practice questions, with answers. The full set is in the app.

### 1. Order the curve-sketching checklist.

**Answer:**

1. Find the domain and the intercepts
2. Find the vertical and horizontal asymptotes
3. Use the sign of f' to find where the curve rises and falls and to locate the critical points
4. Use the sign of f'' to find the concavity and the inflection points
5. Assemble all the features into a sketch

**Why:** The order moves from position to slope to bending: intercepts and asymptotes fix where the curve is, f' gives the up-and-down skeleton, and f'' supplies the shape between the turning points. Sketching from f' alone gets the direction right and the shape wrong.

Page: https://tryals.app/practice/mathematics-i/curve-sketching/order-the-curve-sketching-checklist

### 2. Engineers replace sin θ with θ for small angles, and pendulum clocks work because of it. What is being assumed, and when does it break?

A. That sine is exactly linear near zero, which it is
B. That the pendulum is short, which keeps the angle small automatically
C. That the first Taylor term dominates for small θ, with error like θ³
D. That the angle is measured in degrees rather than radians

**Answer:** C. That the first Taylor term dominates for small θ, with error like θ³

**Why:** The discarded term is θ³/6, so at 0.1 radians the error is about one part in six hundred and the approximation is superb. Push to a wide swing and it fails, a pendulum's period does grow with amplitude, which is exactly why clock escapements are built to keep the arc small.

Page: https://tryals.app/practice/mathematics-i/curve-sketching/engineers-replace-sin-with-for-small-angles-and-pendulum-clocks

### 3. Let $f(x) = (x + 1)/(x - 3)$. Select every TRUE statement about its graph.

A. The x-intercept is at x = -1
B. There is a vertical asymptote at x = 3
C. The horizontal asymptote is the line y = 0
D. The y-intercept is -1/3
E. The horizontal asymptote is the line y = 1

**Answer:** A. The x-intercept is at x = -1; B. There is a vertical asymptote at x = 3; D. The y-intercept is -1/3; E. The horizontal asymptote is the line y = 1

**Why:** f(0) = 1/(-3) = -1/3 and the numerator vanishes at -1. As x grows, (x + 1)/(x - 3) tends to 1, not 0, the numerator and denominator have equal degree, so the ratio of leading coefficients gives the asymptote. Assuming every rational function flattens to y = 0 is the usual error.

Page: https://tryals.app/practice/mathematics-i/curve-sketching/let-f-x-x-1-x-3-select-every-true-statement-about-its

### 4. The linear approximation is exact for a linear function and only approximate for every other.

**Answer:** True

**Why:** True. Taylor's remainder is $\tfrac12 f''(c)(x-a)^2$, which vanishes identically only when the second derivative is zero, that is, for a linear $f$. Everything else carries an error growing with the square of the distance at a rate set by the curvature.

Page: https://tryals.app/practice/mathematics-i/curve-sketching/the-linear-approximation-is-exact-for-a-linear-function-and-only

### 5. Newton's method finds a root by repeatedly stepping along the tangent line: x − f(x)/f'(x). Why does following a straight line converge on the root of a curve?

A. Because near the root the tangent closely tracks the curve.
B. Because f'(x) measures the distance to the root, which the formula then subtracts.
C. Because the method averages successive guesses, and averages converge.
D. Because the tangent line and the curve share the same root, so one step suffices exactly.

**Answer:** A. Because near the root the tangent closely tracks the curve.

**Why:** Each step replaces the curve by its tangent and solves the easy problem of where a line crosses zero, not the true root, but closer, and the next tangent approximates better still. The number of correct digits roughly doubles per step. It is not unconditional: a nearly horizontal tangent throws the guess far away, and a poor start can cycle or find a different root.

Page: https://tryals.app/practice/mathematics-i/curve-sketching/newtons-method-finds-a-root-by-repeatedly-stepping-along-the-tangent

### 6. Sort each function by its symmetry.

**Answer:**

- Even: x squared, x to the fourth - 3 x squared, cos x
- Odd: x cubed, x cubed - x
- Neither: x squared + x

**Why:** Even means f(-x) = f(x), giving symmetry about the vertical axis; odd means f(-x) = -f(x), giving symmetry about the origin. Mixing even and odd powers, as in x squared plus x, destroys both, so most functions are neither, and symmetry is a bonus rather than a rule.

Page: https://tryals.app/practice/mathematics-i/curve-sketching/sort-each-function-by-its-symmetry