Practice Sensitivity (Meta) in Math

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

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Quick 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?

Showing a random 20 of 50 problems.

Example 1

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 2

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 3

hard
For f(x)=1(x−2)2, identify the input region where the output is most sensitive to perturbations.

Example 4

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 5

challenge
For f(x)=sin⁡(1000x), why is computing f at large x in floating point highly sensitive to representation of x?

Example 6

medium
A model predicts exam score S=10log⁡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 7

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 8

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

Example 9

hard
True or false: a small derivative magnitude at a point guarantees that f is insensitive there over a wide neighborhood.

Example 10

easy
For f(x)=x2, compute Δf when x goes from 4 to 4.1.

Example 11

hard
The quadratic formula computes x=−b+b2−4ac2a. For a=1,b=106,c=1, explain why a naive direct computation is highly sensitive to rounding, and give the stable form.

Example 12

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 13

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

Example 14

medium
A retirement formula A=P(1+r)t with P=10000, t=20 years. Compute A for r=0.06 vs r=0.07 to gauge sensitivity to the rate.

Example 15

easy
A measured side s of a cube has 1% error. Volume is s3. Approximately what percent error in volume?

Example 16

easy
Compute f(x)=x3 at x=2 and x=2.1. Find the sensitivity: by what percentage does f change when x changes by 5%?

Example 17

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 18

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 19

medium
In the compound interest formula A=P(1+r)t, compute the sensitivity of A to a small change Δr in the interest rate, using the derivative dAdr.

Example 20

hard
For f(a,b)=a⋅b at (a,b)=(10,2), compute the linearized change if both a and b each increase by 0.1.
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