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Preparation for Calculus

P.4 Fitting Models to Data

Focus on linear data models, exponential data models, residuals and fit.

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Ideas in this lesson

01Linear data modelsWorked steps · five questions02Exponential data modelsWorked steps · five questions03Residuals and fitWorked steps · five questions
Idea 1 of 3

Linear data models

Choose a linear model when changes per unit input are approximately constant. Interpret its parameters in context.

Worked example 1

A quantity begins at 11 and increases by 22 per unit. Find its value after 44 units.
  1. Q(t)=2t+1Q(t) = 2t + 1
  2. Q(4)=2(4)+1=9Q(4) = 2(4) + 1 = 9

Answer: 99

Worked example 2

A quantity begins at 22 and increases by 33 per unit. Its value is 1414. How many units have elapsed?
  1. Q(t)=3t+2Q(t) = 3t + 2
  2. t=14−23=4t = \dfrac{14 - 2}{3} = 4

Answer: 44

Idea 2 of 3

Exponential data models

Choose exponential growth when equal input changes produce approximately equal multiplicative changes.

Worked example 1

A population starts at 22 and doubles each hour. Find P(1)P(1).
  1. P(t)=2⋅2tP(t) = 2 \cdot 2^{t}
  2. P(1)=2⋅2=4P(1) = 2 \cdot 2 = 4

Answer: 44

Worked example 2

A population starts at 33 and doubles each hour. At what time does the population reach 1212?
  1. P(t)=3⋅2tP(t) = 3 \cdot 2^{t}
  2. 2t=4  ⟹  t=22^{t} = 4 \implies t = 2

Answer: 22

Idea 3 of 3

Residuals and fit

A residual is observed minus predicted. Squaring residuals prevents positive and negative errors from canceling.

Worked example 1

Fit y=2x+2y = 2x + 2 to the data below. Find the residual at x=2x = 2.
xx001122
yy224477
  1. ŷ =2,4,6= 2, 4, 6
  2. Residuals y−y - ŷ =0,0,1= 0, 0, 1
  3. Residual at x=2=7−6=1x = 2 = 7 - 6 = 1

Answer: 11

Worked example 2

Fit y=2x+3y = 2x + 3 to the data below. Find the sum of squared residuals.
xx001122
yy335588
  1. ŷ =3,5,7= 3, 5, 7
  2. Residuals y−y - ŷ =0,0,1= 0, 0, 1
  3. SSE=02+02+12=1SSE = 0^{2} + 0^{2} + 1^{2} = 1

Answer: 11

Resource alignment: Larson & Edwards, Calculus of a Single Variable, Tenth Edition, §P.4,pP.4, p. 3131. Explanations, diagrams and numerical practice variants are original MathLab content aligned to this section.