Practice Sensitivity in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

easy
For f(x)=10xf(x)=10x, compute the sensitivity at x=3x=3 with Δx=0.5\Delta x=0.5 and compare to Δx=0.1\Delta x=0.1.

Example 2

easy
For f(x)=xf(x)=x, what is the sensitivity?

Example 3

easy
For f(x)=x2f(x)=x^2, estimate the sensitivity near x=10x=10 using f(10.1)−f(10)0.1\frac{f(10.1)-f(10)}{0.1}.

Example 4

hard
For f(x)=ln⁡xf(x)=\ln x, the sensitivity is 1/x1/x. By what factor is ff more sensitive at x=0.5x=0.5 than at x=50x=50?

Example 5

challenge
For f(x)=x3f(x) = x^3, the sensitivity (slope) is 3x23x^2. Find every input where the sensitivity equals 12, and state where the function is least sensitive.

Example 6

hard
For f(x)=sin⁡xf(x)=\sin x, estimate the slope at x=π/3x=\pi/3 and explain why the function is more sensitive to input near x=0x=0 than near x=π/2x=\pi/2.

Example 7

easy
Between f(x)=20xf(x)=20x and g(x)=4xg(x)=4x, which is more sensitive to input changes?

Example 8

medium
A weather model uses T(P)=0.1P2T(P)=0.1P^2 where PP is pressure. If P=10P=10 and measurement error is ΔP=±0.5\Delta P=\pm0.5, estimate the resulting error in TT.

Example 9

medium
For f(x)=1xf(x)=\frac{1}{x}, compare sensitivity at x=0.5x=0.5 vs x=5x=5, using slope −1/x2-1/x^2.

Example 10

medium
For f(x)=x2f(x)=x^2, find ALL inputs where the sensitivity (slope 2x2x) equals 1414.

Example 11

medium
Two models predict the same output at x=5x = 5, but model A has slope 1 and model B has slope 8 there. If xx is uncertain by ±0.1\pm 0.1, which model's prediction is more reliable?

Example 12

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

Example 13

medium
For f(x)=10xf(x) = 10x, an input measured as 4±0.34 \pm 0.3 produces what output, with what uncertainty?

Example 14

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 15

challenge
A four-stage cascade has local sensitivities 0.5,2,0.8,1.50.5, 2, 0.8, 1.5. Compute the total sensitivity and decide whether the cascade amplifies or attenuates a small input perturbation.

Example 16

hard
Two predictive models for output yy given xx: M1M_1 has ∣y′(x)∣=2.5|y'(x)|=2.5 and M2M_2 has ∣y′(x)∣=0.4|y'(x)|=0.4 at the operating point. If xx is measured with ±0.1\pm 0.1 uncertainty, which model's prediction has lower output uncertainty?

Example 17

medium
An input has a ±0.2\pm 0.2 uncertainty. After f(x)=25xf(x) = 25x, what is the approximate uncertainty in the output?

Example 18

medium
For f(x)=2xf(x)=2x followed by g(u)=5u+1g(u)=5u+1, what is the overall sensitivity of g(f(x))g(f(x)) to xx?

Example 19

easy
For f(x)=3x+7f(x) = 3x + 7, what is the sensitivity of the output to the input (change in output per unit change in input)?

Example 20

medium
For f(x)=x2f(x) = x^2, compare the sensitivity near x=1x = 1 with the sensitivity near x=5x = 5 (use the derivative 2x2x). Which input region is more sensitive?
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Rate of ChangeDerivative