Sensitivity Examples in Math: 45 Problems with Answers

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

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

Concept Recap

In the context of functions, sensitivity measures how much the output changes in response to a small change in the input — high sensitivity means small input changes cause large output changes.

A sensitive scale notices tiny weight differences. An insensitive one doesn't.

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 measures how much the output moves for a tiny change in the input — steep means very sensitive.

Common stuck point: The procedure for sensitivity is the easy part; the trap is calling a function 'sensitive' globally. Asking "Does a small change in the input produce a large change in the output here?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does a small change in the input produce a large change in the output here?

Worked Examples

Example 1

easy

Compute the sensitivity

\Delta F / \Delta x

for

F(x) = 3x^2

x = 5

with perturbation

\Delta x = 0.1

Answer

Sensitivity

\approx 30.3

; derivative

F'(5)=30

First step

Compute

F(5) = 3(25)=75

and

F(5.1)=3(26.01)=78.03

Full solution

2
$\Delta F = 78.03-75=3.03$ . Sensitivity $= \Delta F / \Delta x = 3.03/0.1 = 30.3$ .
3
Compare to derivative: $F'(x)=6x$ , so $F'(5)=30$ . The sensitivity approximates the derivative, with a small discrepancy due to $\Delta x$ being finite.

Sensitivity

\Delta F/\Delta x

measures how much the output changes per unit change in input. As

\Delta x\to0

, this ratio converges to the derivative. High sensitivity means small input changes produce large output changes.

Example 2

hard

A function

F(x)=e^x

is highly sensitive near large

x

. Compare the sensitivity at

x=0

and

x=5

using a perturbation of

\Delta x=0.01

Example 3

medium

A two-stage signal: stage A scales by

0.4

, stage B scales by

7

. Find the total sensitivity factor and decide whether a small input error grows or shrinks.

Example 4

hard

For

f(x)=\sin x

, estimate the slope at

x=\pi/3

and explain why the function is more sensitive to input near

x=0

than near

x=\pi/2

Example 5

challenge

A four-stage cascade has local sensitivities

0.5, 2, 0.8, 1.5

. Compute the total sensitivity and decide whether the cascade amplifies or attenuates a small input perturbation.

Practice Problems

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

Example 1

easy

For

f(x)=10x

, compute the sensitivity at

x=3

with

\Delta x=0.5

and compare to

\Delta x=0.1

Example 2

medium

A weather model uses

T(P)=0.1P^2

where

P

is pressure. If

P=10

and measurement error is

\Delta P=\pm0.5

, estimate the resulting error in

T

Example 3

easy

For the linear function

f(x) = 5x

, by how much does the output change when the input increases by 1?

Example 4

easy

Which function is more sensitive to its input near

x = 1

f(x) = 2x

g(x) = 10x

Example 5

easy

A scale shows 100.0 g for a true 100 g object and 100.5 g when 0.5 g is added. Is this scale sensitive to small changes?

Example 6

easy

For

f(x) = 3x + 7

, what is the sensitivity of the output to the input (change in output per unit change in input)?

Example 7

easy

Does doubling the input of

f(x) = 4x

double the output? Use this to confirm its sensitivity is constant.

Example 8

easy

True or false: a large output value automatically means the function is highly sensitive at that point.

Example 9

easy

For

f(x) = 100x

measuring dollars per hour, what does the sensitivity 100 mean in words?

Example 10

easy

Between

f(x) = x

and

g(x) = 0.1x

, which output is LESS sensitive to input changes?

Example 11

medium

For

f(x) = x^2

, estimate the sensitivity near

x = 3

using

f(3.1) - f(3)

divided by

0.1

Example 12

medium

For

f(x) = x^2

, compare the sensitivity near

x = 1

with the sensitivity near

x = 5

(use the derivative

2x

). Which input region is more sensitive?

Example 13

medium

A measurement passes through

f(x) = 2x

then

g(u) = 3u

. By what factor does a small input error get amplified by the chain

g(f(x))

Example 14

medium

An input has a

\pm 0.2

uncertainty. After

f(x) = 25x

, what is the approximate uncertainty in the output?

Example 15

medium

Two scales: scale A reads

1000x

grams-units, scale B reads

2x

grams-units, for the same input

x

in grams. Which is more sensitive, and why is the larger reading alone not the deciding factor?

Example 16

medium

For

f(x) = \sqrt{x}

, is the output more sensitive to input near

x = 0.01

or near

x = 100

? (Slope is

\frac{1}{2\sqrt{x}}

Example 17

medium

A model output is

y = 4a + 9b

. Which input parameter is the output more sensitive to, and by what ratio?

Example 18

challenge

A signal passes through three stages with local sensitivities

1.5

0.4

, and

5

. Does a small input error grow or shrink overall, and by what factor?

Example 19

challenge

For

f(x) = x^3

, the sensitivity (slope) is

3x^2

. Find every input where the sensitivity equals 12, and state where the function is least sensitive.

Example 20

challenge

A bank balance is

B = 1000(1.05)^t

. Its sensitivity to

t

grows over time. Without calculus, compare the one-year change from year 0 to year 1 with the change from year 10 to year 11, as a ratio.

Example 21

medium

For

f(x) = 10x

, an input measured as

4 \pm 0.3

produces what output, with what uncertainty?

Example 22

medium

Two models predict the same output at

x = 5

, but model A has slope 1 and model B has slope 8 there. If

x

is uncertain by

\pm 0.1

, which model's prediction is more reliable?

Example 23

easy

For

f(x)=7x-2

, what is the sensitivity of

f

x

Example 24

easy

For

f(x)=x^2

, estimate the sensitivity near

x=10

using

\frac{f(10.1)-f(10)}{0.1}

Example 25

easy

For

f(x)=3x+10

, what does the constant

10

tell you about sensitivity? Does it affect the slope?

Example 26

easy

True or false: doubling the input always doubles the sensitivity for a linear function.

Example 27

medium

For

f(x)=x^3

x=2

, estimate the sensitivity using

f(2.1)

and

f(2)

Example 28

medium

For

f(x)=2x

followed by

g(u)=5u+1

, what is the overall sensitivity of

g(f(x))

x

Example 29

medium

An input

x=8

has uncertainty

\pm 0.4

. After

f(x)=15x-3

, what is the output and its approximate uncertainty?

Example 30

medium

For

f(x)=\sqrt{x}

near

x=64

, estimate sensitivity using the slope formula

1/(2\sqrt x)

Example 31

medium

For

f(x)=x^2

, find ALL inputs where the sensitivity (slope

2x

) equals

14

Example 32

medium

For

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

, compare sensitivity at

x=0.5

x=5

, using slope

-1/x^2

Example 33

medium

An input

x=12 \pm 0.5

is fed into

f(x)=4x-9

. State the output and its approximate uncertainty.

Example 34

medium

For

f(x)=2^x

, where is the function more sensitive: near

x=0

or near

x=10

Example 35

hard

For

f(x)=x^3-12x

, find the input where

f

is least sensitive (slope smallest in magnitude).

Example 36

hard

For

f(x)=\ln x

, the sensitivity is

1/x

. By what factor is

f

more sensitive at

x=0.5

than at

x=50

Example 37

hard

Two predictive models for output

y

given

x

M_1

has

|y'(x)|=2.5

and

M_2

has

|y'(x)|=0.4

at the operating point. If

x

is measured with

\pm 0.1

uncertainty, which model's prediction has lower output uncertainty?

Example 38

medium

For

f(x)=4x^2

, find the sensitivity at

x=3

using the derivative.

Example 39

challenge

For

y = x^a

with

a > 0

, find the value of

x

where the elasticity

E = (x/y)\cdot y'

equals 1 in general for arbitrary

a

— and determine for which

a

the elasticity is always

1

Example 40

challenge

For

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

, compute the sensitivity at

x=0

and at

x=10

, and decide which input has the more robust output.

← Back to Sensitivity Formula Explained Practice Mode

Related Concepts

Rate of Change Derivative

Background Knowledge

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

rate of change