Sensitivity (Meta) Examples: 45 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Sensitivity (Meta).

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The degree to which a result or output changes in response to small changes in its inputs, parameters, or assumptions.

Is this result stable, or does a tiny change blow everything up?

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Sensitivity is the ratio of how much the output changes to how much the input changed.

Common stuck point: The procedure for sensitivity (meta) is the easy part; the trap is reporting the raw output change as sensitivity. Asking "Am I measuring how much the output moves PER unit change in the input?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I measuring how much the output moves PER unit change in the input?

Worked Examples

Example 1

easy

Compute

f(x) = x^3

x = 2

and

x = 2.1

. Find the sensitivity: by what percentage does

f

change when

x

changes by 5%?

Answer

5% change in x \Rightarrow \approx 15.8 % change in f

First step

f(2) = 8

f(2.1) = 9.261

Full solution

2
Change in $f$ : $9.261 - 8 = 1.261$ . Relative change in $f$ : $\frac{1.261}{8} \approx 15.8\%$ .
3
Change in $x$ : $\frac{0.1}{2} = 5\%$ .
4
Sensitivity: a 5% increase in $x$ causes about a 15.8% increase in $f$ . The function is sensitive — it amplifies errors by a factor of about 3.

Sensitivity measures how much the output changes relative to the input. For

f(x) = x^3

, the exponent 3 amplifies percentage changes roughly threefold, which can be derived from the derivative:

f'(x)/f(x) \approx 3 \Delta x/x

Example 2

medium

In the compound interest formula

A = P(1+r)^t

, compute the sensitivity of

A

to a small change

\Delta r

in the interest rate, using the derivative

\frac{dA}{dr}

Example 3

medium

A retirement formula

A=P(1+r)^t

with

P=10000

t=20

years. Compute

A

for

r=0.06

r=0.07

to gauge sensitivity to the rate.

Example 4

hard

The quadratic formula computes

x=\frac{-b+\sqrt{b^2-4ac}}{2a}

. For

a=1, b=10^6, c=1

, explain why a naive direct computation is highly sensitive to rounding, and give the stable form.

Example 5

challenge

Heron's formula for triangle area is sensitive when the triangle is thin (near degenerate). For sides

a=1

b=1

c=1.9999

, compute area with both the standard and a numerically stable form and discuss the gap.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

For

f(x)=\sqrt{x}

, compute

f(100)

and

f(104)

. By what percentage does

f

change when

x

increases by 4%?

Example 2

medium

A model predicts exam score

S = 10\log_{10}(h)

where

h

is hours of study. If

h

changes from 10 to 11 hours, compute the change in

S

and discuss sensitivity.

Example 3

easy

A function is f(x) = 1000x. If x changes by 0.01, by how much does f change?

Example 4

easy

Function g(x) = 0.001x. If x changes by 100, by how much does g change? Is g sensitive to x?

Example 5

easy

Computing 1/x near x = 0.001, then near x = 1000. Where is 1/x more sensitive to a small change in x?

Example 6

easy

A measured side s of a square has 1% error. The area is s^2. Approximately what percent error appears in the area?

Example 7

easy

In y = 5x + 2, which input change causes a bigger output change: changing x by 1 or changing the constant 2 by 1?

Example 8

easy

A bank balance grows by interest rate r. A 0.1% change in r barely changes one month's interest but compounds over 30 years. This shows sensitivity can depend on what?

Example 9

easy

Two recipes: A's taste is ruined if salt is off by a pinch; B tolerates large salt swings. Which is more sensitive to the salt input?

Example 10

easy

A formula's answer changes from 5.00 to 5.01 when an input changes by 50%. Is the answer sensitive to that input?

Example 11

medium

For f(x) = x^2 at x = 3, estimate the output change when x increases by 0.1 using the derivative, and compare with the exact change.

Example 12

medium

A system solves A x = b. The condition number of A is 1000. If b has 0.1% relative error, what is the worst-case relative error in x?

Example 13

medium

Profit P = (price - 10) * (1000 - 20*price). Determine whether profit is more sensitive to price near price = 20 or near price = 40, using the slope.

Example 14

medium

A computation evaluates (1 + x) - 1 for x = 1e-16 in floating point and gets 0 instead of 1e-16. Which input region is this operation sensitive to, and what causes the failure?

Example 15

medium

Compound interest: A = P(1 + r)^t. With P = 1000, t = 30, compare final amounts for r = 0.05 and r = 0.06 to gauge sensitivity to r.

Example 16

medium

A formula y = a/(a - b). For a = 100, compare sensitivity of y to a small change in b when b = 1 versus b = 99.

Example 17

challenge

For f(x, y) = x*y at (x, y) = (4, 5), use partial derivatives to find which variable the output is more sensitive to, and the first-order output change if both increase by 0.1.

Example 18

challenge

A relative-sensitivity (elasticity) is defined as E = (x/f) * df/dx. Compute the elasticity of f(x) = x^n and interpret the result.

Example 19

challenge

A root-finding iteration for f(x) = 0 near a root r has error multiplied by |f'(r)|^(-1) per measurement error in f. If f(x) = (x - 2)^3, explain why the root x = 2 is highly sensitive to perturbations in f.

Example 20

medium

A thermostat formula output = k*(target - current) with k = 50. If the temperature reading is off by 0.2 degrees, how much does the output change, and is the system sensitive to reading errors?

Example 21

medium

For f(x) = ln(x), is the output more sensitive to a small change in x near x = 0.1 or near x = 10? Use f'(x) = 1/x.

Example 22

medium

A recipe scales servings by a factor s, and total cost is C(s) = 5s. A spreadsheet rounds s to the nearest 0.5. If true s = 2.2 rounds to 2.0, what is the resulting cost error, and is cost sensitive to this rounding?

Example 23

easy

For

f(x)=20x

, by how much does

f

change when

x

increases by

0.25

Example 24

easy

For

f(x)=x^2

, compute

\Delta f

when

x

goes from

4

4.1

Example 25

easy

A measured side

s

of a cube has

1\%

error. Volume is

s^3

. Approximately what percent error in volume?

Example 26

easy

For

f(x)=\frac{1}{x}

, where is

f

more sensitive to small changes in

x

: near

x=0.1

or near

x=10

Example 27

medium

For

f(x)=\sqrt{x}

, estimate the change in

f

when

x

goes from

25

26

using the derivative.

Example 28

medium

A linear system

Ax=b

has condition number

\kappa(A)=500

. If

b

is known to

0.05\%

relative error, what is the worst-case relative error in

x

Example 29

medium

For

f(x)=\tan x

, compare sensitivity at

x=0

x

near

\pi/2

(e.g.,

x=1.5

Example 30

medium

Output

y=3a^2 b

. If

a

changes by

1\%

and

b

stays fixed, approximately what percent change in

y

Example 31

medium

A model uses

T=10\sqrt{P}

. Compute the sensitivity

dT/dP

P=100

and the approximate change in

T

P

shifts by

1

Example 32

medium

Cancellation: computing

\sqrt{x+1}-\sqrt{x}

for large

x

loses precision. Rewrite to reduce sensitivity to rounding.

Example 33

medium

A function

f(x,y)=x^2+y

(3,4)

. Which input has greater first-order sensitivity?

Example 34

hard

For

f(x)=\frac{1}{(x-2)^2}

, identify the input region where the output is most sensitive to perturbations.

Example 35

hard

For

f(a,b)=a\cdot b

(a,b)=(10,2)

, compute the linearized change if both

a

and

b

each increase by

0.1

Example 36

hard

The elasticity of

f(x)=x^n

is defined as

E=\frac{x}{f}\cdot f'(x)

. Compute

E

for

f(x)=5x^{-2}

Example 37

hard

A bridge load model

L=10000+50w-2w^2

where

w

is wind speed. Find the wind speed where

L

is most sensitive (in magnitude) to

w

over

0\le w\le 30

Example 38

medium

In a chain

f(g(x))

with

f'=4

and

g'=0.5

at the relevant points, by what factor does input error get amplified?

Example 39

challenge

For a 2x2 matrix

A=\begin{pmatrix}1&1\\ 1&1.01\end{pmatrix}

, the system

Ax=b

is ill-conditioned. Compute

\det A

and explain why solving

Ax=b

is sensitive to small changes in

b

Example 40

challenge

For

f(x)=\sin(1000x)

, why is computing

f

at large

x

in floating point highly sensitive to representation of

x

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Related Concepts

Local vs Global Behavior Robustness

Background Knowledge

These ideas may be useful before you work through the harder examples.

local vs global behavior