Practice Sensitivity (Meta) in Math

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

Worksheet PDF Free Answer-key PDF Family

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(xโˆ’2)2f(x)=\frac{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)=sinโก(1000x)f(x)=\sin(1000x), why is computing ff at large xx in floating point highly sensitive to representation of xx?

Example 6

medium
A model predicts exam score S=10logโก10(h)S = 10\log_{10}(h) where hh is hours of study. If hh changes from 10 to 11 hours, compute the change in SS 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 ff is insensitive there over a wide neighborhood.

Example 10

easy
For f(x)=x2f(x)=x^2, compute ฮ”f\Delta f when xx goes from 44 to 4.14.1.

Example 11

hard
The quadratic formula computes x=โˆ’b+b2โˆ’4ac2ax=\frac{-b+\sqrt{b^2-4ac}}{2a}. For a=1,b=106,c=1a=1, b=10^6, 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)tA=P(1+r)^t with P=10000P=10000, t=20t=20 years. Compute AA for r=0.06r=0.06 vs r=0.07r=0.07 to gauge sensitivity to the rate.

Example 15

easy
A measured side ss of a cube has 1%1\% error. Volume is s3s^3. Approximately what percent error in volume?

Example 16

easy
Compute f(x)=x3f(x) = x^3 at x=2x = 2 and x=2.1x = 2.1. Find the sensitivity: by what percentage does ff change when xx 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)tA = P(1+r)^t, compute the sensitivity of AA to a small change ฮ”r\Delta r in the interest rate, using the derivative dAdr\frac{dA}{dr}.

Example 20

hard
For f(a,b)=aโ‹…bf(a,b)=a\cdot b at (a,b)=(10,2)(a,b)=(10,2), compute the linearized change if both aa and bb each increase by 0.10.1.
โ† Back to Sensitivity (Meta) Worked Examples Formula Explained