Complete Theory

Worked Examples

Example 1Gradient and tangent plane

Example 2A limit that does not exist

Example 3Critical points

Example 4Implicit function (Dini)

Example 5Constrained optimisation (Lagrange)

Exercise 1GradientMedium

📋 Problem to solve

Compute

\nabla f(1,-1)

for

f=xe^y+y^2

Exercise 2Directional derivativeMedium

📋 Problem to solve

Derivative of

f=x^2+y^2

(1,1)

along

\hat v=(1,0)

Exercise 3LimitsHard

📋 Problem to solve

Decide whether

\lim_{(x,y)\to(0,0)}\dfrac{x^2y}{x^4+y^2}

exists.

Exercise 4TaylorMedium

📋 Problem to solve

Second-order Taylor of

e^{x+y}

(0,0)

Exercise 5Implicit functionHard

📋 Problem to solve

For

x^2+y^2+z^2=1

find

\partial z/\partial x

(1/\sqrt3,1/\sqrt3,1/\sqrt3)

Exercise 6LagrangeHard

📋 Problem to solve

Minimise

f=x^2+y^2

on the constraint

x+2y=5

📚

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