Differentials. Approximations Introduction - Part 1
Almaz Khusnutdinov•Math enthusiast, love coding and mathematics
Published 8/22/2025
Problem 1.1 - Linear approximation and error
Estimate 50 using the linear approximation of f(x)=x at a=49. Then give the absolute error compared to 52.Detailed solution.For f(x)=x1/2,f′(x)=2x1.Choose a=49(49=7) and Δx=50−49=1.By linearization, f(a+Δx)≈f(a)+f′(a)Δx.Δy≈f′(49)Δx=2⋅71⋅1=141.∴50=f(50)≈f(49)+Δy=7+141=1499≈7.071428571.The exact value is 52≈7.071067812.Absolute error =1499−52≈3.6076×10−4.Answer: 50≈1499≈7.07143,error≈3.61×10−4.
Problem 1.2 - Differentials for volume of a sphere.
Let V(r)=34πr3.Use differentials to estimate the increase in volume when r:10→10.2cm.Also estimate the relative change VΔV.Answer: ΔV≈80πcm3(≈251.33cm3),VΔV≈0.06(6%).
Affine Approximation
Problem 2.1 - Proof - Tangent-line approximation is first order
Let f be differentiable at a.Show that L(x)=f(a)+f′(a)(x−a) is the unique affine function with f(x)=L(x)+o(x−a)as x→a.Answer (Outline): 1)Define L(x)=f(a)+f′(a)(x−a).Thenx−af(x)−L(x)=x−af(x)−f(a)−f′(a)x→a0,so f(x)−L(x)=o(x−a).2)Uniqueness: If A(x)=α+β(x−a) also satisfies f(x)−A(x)=o(x−a),thenx→alim(x−af(x)−A(x))=0⇒β=x→alimx−af(x)−f(a)=f′(a).Evaluating at x=a gives α=f(a).Hence A(x)=L(x) and uniqueness holds.
Problem 2.2 - Affine Approximation.
Let f(x)=ln(1+x). (a) Find the unique affine map A(x)=α+βx such that f(x)=A(x)+o(x) as x→0.(b) Use this affine approximation to estimate ln(1.1).Answer: α=0,β=1,A(x)=x;ln(1.1)≈0.1.
Problem 2.3 - Chain Rule via Affine.
Let f be differentiable at a,and g be differentiable at f(a). Using the first-order affine approximation property, prove g∘f is differentiable at a and (g∘f)′(a)=g′(f(a))f′(a).Answer (Outline): Write f(x)=f(a)+f′(a)(x−a)+o(x−a). Let u=f(x). Then g(u)=g(f(a))+g′(f(a))(u−f(a))+o(u−f(a)).Substitute u−f(a)=f′(a)(x−a)+o(x−a) to obtain g(f(x))=g(f(a))+g′(f(a))f′(a)(x−a)+o(x−a).Thus (g∘f)′(a)=g′(f(a))f′(a).
Problem 2.4 - Derivative of the Inverse via Affine Approximation
Suppose f is differentiable at a,f′(a)=0,and is invertible near a. Let b=f(a). Using the affine approximation property, show (f−1) is differentiable at b with (f−1)′(b)=f′(a)1.Answer (Outline): Write f(x)=b+f′(a)(x−a)+o(x−a). Set y=f(x). Solve for x in terms of y:x=a+f′(a)1(y−b)+o(y−b).Hence f−1(y)=a+f′(a)1(y−b)+o(y−b).Therefore (f−1)′(b)=1/f′(a).
Problem 2.5 - Affine Approximation #2
Let f(x)=e2xsinx. (a) Find the unique affine map A(x)=α+βxsuch that f(x)=A(x)+o(x)as x→0.(b) Use this affine approximation to estimate e0.2sin(0.1).Answer: α=0,β=1,A(x)=x;e0.2sin(0.1)≈0.1.
Problem 2.6 - Affine Approximation #3
Let f(x)=1+3x+x2. (a) Find the unique affine map A(x)=α+βxsuch that f(x)=A(x)+o(x)as x→0.(b) Use this affine approximation to estimate 1.1525.Answer: α=1,β=23,A(x)=1+23x;1.1525≈1.075.
Problem 2.7 - Affine Approximation #4
Let f(x)=ln(2+x). (a) Find the unique affine map A(x)=α+βxsuch that f(x)=A(x)+o(x−1)as x→1.(b) Use this affine approximation to estimate ln(3.1).Answer: α=ln3−31,β=31,A(x)=(ln3−31)+31x;ln(3.1)≈ln3+301≈1.13195.
Problem 2.8 - Affine Proof #3
Let f,g be differentiable at a. Using the first-order affine approximation property, prove (fg) is differentiable at a and (fg)′(a)=f′(a)g(a)+f(a)g′(a).Answer (Outline): f(x)=f(a)+f′(a)(x−a)+o(x−a),g(x)=g(a)+g′(a)(x−a)+o(x−a).Multiply: f(x)g(x)=f(a)g(a)+(f′(a)g(a)+f(a)g′(a))(x−a)+o(x−a),since (x−a)o(x−a)=o(x−a) and o(x−a)⋅o(x−a)=o(x−a). Hence (fg)′(a)=f′(a)g(a)+f(a)g′(a).
Problem 2.9 - Affine Proof #4
Let f be differentiable at a,g be differentiable at a,and g(a)=0. Using affine approximation, prove (gf) is differentiable at a and (gf)′(a)=g(a)2f′(a)g(a)−f(a)g′(a).Answer (Outline): f(x)=f(a)+f′(a)(x−a)+o(x−a),g(x)=g(a)+g′(a)(x−a)+o(x−a).Seek h(x) with g(x)h(x)=1+o(x−a). Take h(x)=g(a)1−g(a)2g′(a)(x−a)+o(x−a).Then g(x)f(x)=f(x)h(x)=g(a)f(a)+g(a)2f′(a)g(a)−f(a)g′(a)(x−a)+o(x−a).Therefore (gf)′(a)=g(a)2f′(a)g(a)−f(a)g′(a).
Problem 2.10 - Affine Approximation #5
Let f(x)=ln(1+2x+x2). (a) Find the unique affine map A(x)=α+βxsuch that f(x)=A(x)+o(x)as x→0.(b) Use this affine approximation to estimate ln(1.21).Answer: α=0,β=2,A(x)=2x;ln(1.21)≈0.200.Let f(x)=(1+3x)1/3. (a) Find the unique affine map A(x)=α+βxsuch that f(x)=A(x)+o(x)as x→0.(b) Use this affine approximation to estimate 31.06.Answer: α=1,β=1,A(x)=1+x;31.06≈1.020.Let f(x)=1+xsin(2x). (a) Find the unique affine map A(x)=α+βxsuch that f(x)=A(x)+o(x)as x→0.(b) Use this affine approximation to estimate 1.1sin(0.2).Answer: α=0,β=2,A(x)=2x;1.1sin(0.2)≈0.200.
Problem 2.11 - Affine Approximation
Let f(x)=1−xex. (a) Find the unique affine map A(x)=α+βxsuch that f(x)=A(x)+o(x)as x→0.(b) Use this affine approximation to estimate 0.95e0.05.Answer: α=1,β=23,A(x)=1+23x;0.95e0.05≈1.075.Let f(x)=(1+x)(1+2x)=1+3x+2x2. (a) Find the unique affine map A(x)=α+βxsuch that f(x)=A(x)+o(x)as x→0.(b) Use this affine approximation to estimate 1.32.Answer: α=1,β=23,A(x)=1+23x;1.32≈1.150.
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