# Taylor Approximation

Mathematics I · Differential Calculus · https://tryals.app/learn/mathematics-i/taylor-approximation

Complicated functions like $\sin x$ or $e^x$ can be approximated near a point by **polynomials** built from their derivatives. Since polynomials use only basic arithmetic, this is how computers actually evaluate them.

The simplest case is the **linear approximation**:
$$f(x) \approx f(a) + f'(a)(x - a)$$
Here, $f(a)$ is the value, $f'(a)$ is the slope, and $(x-a)$ is the displacement from center $a$.

The general **Taylor polynomial** improves accuracy by adding higher-derivative terms:
$$f(x) \approx f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3$$
When the expansion center is $a = 0$, it is specifically called a **Maclaurin series**.

## Terms & Pitfalls

Each term in a Taylor polynomial matches a specific geometric feature of the function at the center point $a$.

| Term | Matches | Geometric Meaning |
| --- | --- | --- |
| Constant | $f(a)$ | Initial **value** |
| Linear | $f'(a)$ | Initial **slope** |
| Quadratic | $f''(a)$ | Initial **curvature** |

| Feature | Description |
| --- | --- |
| **Locality** | Accuracy degrades as $|x-a|$ grows. |
| **Common Pitfall** | Expecting a truncated polynomial to hold **far** from $a$. A few terms are great nearby but can be wildly wrong far away. |

**More terms** generally widen the region of accuracy, but a finite polynomial is always a local approximation.

## Practice questions

4 of this lesson's 10 practice questions, with answers. The full set is in the app.

### 1. Order the terms of the Taylor polynomial of f about a, from lowest degree to highest.

**Answer:**

1. f(a)
2. f'(a)(x - a)
3. f''(a)(x - a) squared, divided by 2 factorial
4. f'''(a)(x - a) cubed, divided by 3 factorial

**Why:** The constant fixes the value, the linear term fixes the slope, the quadratic fixes the curvature, and so on, each term corrects a feature the previous ones could not capture. That correspondence between degree and derivative order is the whole design of the series.

Page: https://tryals.app/practice/mathematics-i/taylor-approximation/order-the-terms-of-the-taylor-polynomial-of-f-about-a-from-lowest

### 2. A speed camera pair times a car between two points and issues a ticket for exceeding the limit *somewhere* between them, without observing the speedometer. Which theorem licenses that?

A. The Intermediate Value Theorem, since speed passes through every intermediate value
B. The Fundamental Theorem of Calculus, which relates distance to speed
C. None, the inference is legally accepted but mathematically unsound
D. The Mean Value Theorem, so the average is attained at some instant

**Answer:** D. The Mean Value Theorem, so the average is attained at some instant

**Why:** Average speed must be attained instantaneously at some point in between, so the ticket follows from the arithmetic without anyone seeing the dial. Average-speed enforcement is a mathematical theorem installed on a motorway, one of the few places the Mean Value Theorem is applied to you directly.

Page: https://tryals.app/practice/mathematics-i/taylor-approximation/a-speed-camera-pair-times-a-car-between-two-points-and-issues-a

### 3. Select every TRUE statement about Taylor approximation.

A. The approximation is most accurate near the centre a
B. Adding more terms generally improves accuracy
C. A Maclaurin series is a Taylor series centred at a = 0
D. A finite Taylor polynomial matches f exactly everywhere
E. The linear approximation is the tangent line at a

**Answer:** A. The approximation is most accurate near the centre a; B. Adding more terms generally improves accuracy; C. A Maclaurin series is a Taylor series centred at a = 0; E. The linear approximation is the tangent line at a

**Why:** Taylor approximation is fundamentally LOCAL: accurate near a, degrading as the distance grows, improvable by more terms but never exact for a finite polynomial. The exponential grows faster than any polynomial, so no truncation can match it everywhere.

Page: https://tryals.app/practice/mathematics-i/taylor-approximation/select-every-true-statement-about-taylor-approximation

### 4. The Mean Value Theorem says a differentiable function on [a,b] has some point where the instantaneous rate equals the average rate over the whole interval. Why is that guaranteed rather than merely plausible?

A. Because the derivative is defined as an average rate, so the two are the same quantity.
B. Because the theorem assumes the function is linear, in which case the rates are equal everywhere.
C. Because always below could not climb far enough, always above too far.
D. Because averages are always attained by continuous quantities, which is a general property of averages.

**Answer:** C. Because always below could not climb far enough, always above too far.

**Why:** The driving version makes it obvious: cover 120 miles in two hours and if the speedometer never read 60, it was either always below and you fell short, or always above and you overshot. The hypotheses do real work, drop differentiability and |x| on [−1, 1] is a counterexample. It underlies f' > 0 implying increasing, and the error bounds on Taylor approximations.

Page: https://tryals.app/practice/mathematics-i/taylor-approximation/the-mean-value-theorem-says-a-differentiable-function-on-a-b-has