Problems · LeoMath

Few but real exercises. The goal is not drilling, but checking whether you truly understood the definition.

Foundations of calculus

Limits

Compute

\displaystyle\lim_{x\to 0}\frac{\sin 3x}{x}

Let

f(x)=2x+1

. Which

\delta

guarantees

|f(x)-3|<\varepsilon

whenever

0<|x-1|<\delta

The derivative

Let

f(x)=x^3-2x

. Find

f'(2)

Compute

\displaystyle\lim_{h\to0}\frac{2^h-1}{h}

to three decimals. This is the derivative of

2^x

x=0

Mean value theorem

For

f(x)=x^3

[0,2]

, find

\xi\in(0,2)

with

f'(\xi)=\dfrac{f(2)-f(0)}{2-0}

(three decimals).

For

f(x)=|x|

[-1,1]

the secant slope is

0

, yet no point has derivative

0

. Which hypothesis of the mean value theorem fails?

The integral

Compute

\displaystyle\int_0^1 x^2\,dx

to four decimals.

Let

F(x)=\displaystyle\int_0^x e^{-t^2}\,dt

. Then

F'(x)

equals

Taylor expansion

In the Taylor series of

\sin x

0

, what is the coefficient of

x^3

(four decimals)?

Approximating

\cos x

1-\tfrac{x^2}{2}

for

|x|\le0.5

, the Lagrange remainder bounds the error by approximately

Foundations of linear algebra

Vectors

What is the dimension of the subspace of

\mathbb R^3

spanned by

(1,0,1),(0,1,1),(1,1,2)

Under the usual operations, which of these is not a vector space?

Inner products & Cauchy–Schwarz

Find the cosine of the angle between

u=(1,2,2)

and

v=(2,-1,2)

(three decimals).

For reals

a_1,\dots,a_n

, Cauchy–Schwarz gives

(a_1+\cdots+a_n)^2\le C\,(a_1^2+\cdots+a_n^2)

. The smallest such

C

Linear maps

A linear map

T:\mathbb R^2\to\mathbb R^2

has

T(1,0)=(2,1)

and

T(0,1)=(0,3)

. Find the second coordinate of

T(1,2)

Which of the following maps is linear?

Matrices

Compute

\det\begin{pmatrix}2&1\\3&4\end{pmatrix}

The matrix (in the standard basis) of the counter-clockwise rotation by

90^\circ

Eigenvalues

Find the largest eigenvalue of

\begin{pmatrix}2&1\\1&2\end{pmatrix}

The eigenvectors of the shear

\begin{pmatrix}1&1\\0&1\end{pmatrix}

are

Introduction to differential equations

What is an ODE

Solve

y'=2y,\ y(0)=3

and give

y(1)

to two decimals.

What is the order of

y'''+y\,y'=\sin t

First-order equations

Solve

y'=y^2,\ y(0)=1

and find

y(0.5)

Which integrating factor solves

y'+2y=e^t

Second-order linear equations

What is the larger root of the characteristic equation of

y''-3y'+2y=0

The general solution of

y''+4y=0

Numerical solutions

Apply Euler's method with

h=0.5

y'=y,\ y(0)=1

for two steps. What approximation of

y(1)

results?

RK4 has global error

O(h^4)

. Halving the step size divides the error by roughly