Maths Weeks 1-3

What is the derivative of x^n according to the power rule?

The derivative of x^n is n*x^(n-1).

What is Leibniz's rule for derivatives?

The derivative of a product of functions can be derived by taking the sum of the derivative of each function while holding the other function constant.

How do you differentiate the product of two functions, such as x^2 and sin(x)?

Use the product rule: (d/dx)(x^2 * sin(x)) = (d/dx)(x^2) * sin(x) + x^2 * (d/dx)(sin(x)).

How do you differentiate a quotient when rewriting it as a sum of powers?

Rewrite the quotient and then differentiate each term as a power, using the power rule.

What are the values for which the tangent function is undefined?

The tangent function is undefined where the cosine function is zero, specifically at (2n+1)(π/2) for any integer n.

What is the derivative of the tangent function?

The derivative of the tangent function is sec^2(x) or 1 + tan^2(x).

What is the chain rule in differentiation?

The chain rule states that to differentiate a composite function, derive the outer function and multiply it by the derivative of the inner function.

What is the process to differentiate the function sin(g(x))?

First differentiate sin(u) to get cos(u), then multiply by the derivative of g(x).

When is a function defined as differentiable at a point?

A function is differentiable at a point if the limit of the difference quotient exists at that point.

What approach do we use to check if a piecewise function is differentiable at a point where it is defined differently?

Use the definition of the derivative, computing the limit of the difference quotient as the variable approaches that point.

What does the squeeze theorem help determine?

The squeeze theorem helps establish the limit of a function by bounding it between two other functions that converge to the same limit.

How would you denote the derivative of the function h that is defined as the composition of two functions f and g?

h'(x) = f'(g(x)) * g'(x) by the chain rule.

What is the formal definition of the derivative at a point?

f'(c) = lim (h -> 0) [(f(c + h) - f(c)) / h], if this limit exists.

What must you check when determining if g(f(x)) is differentiable at a certain point?

You must ensure that both f and g are differentiable at their respective points, and that g's output is applicable as input for f.

What can be said about the derivative of x * sin(1/x) at x = 0?

The derivative does not exist at x = 0 since it results in an oscillation as x approaches zero.

What is the derivative of a function that is defined piecewise and has different definitions for different ranges of x?

You need to separately analyze each piece and verify the limits at the points of change to ensure continuity and differentiability.

What does the Intermediate Value Theorem state about continuous functions?

If a function f is continuous on a closed interval [a, b] and takes on values f(a) and f(b), then it must take on all intermediate values between f(a) and f(b) at some point c in (a, b).

What is required for the Intermediate Value Theorem to apply?

The function must be continuous on a closed interval and the endpoints of the interval must take on opposite signs.

How does the Intermediate Value Theorem demonstrate the existence of roots in equations?

If f(a) < 0 and f(b) > 0, there is at least one c in (a, b) such that f(c) = 0.

What is the relationship between continuity and maximum/minimum values on an interval?

A continuous function on a closed interval will always have both a maximum and minimum value.

In the context of solving equations using the Intermediate Value Theorem, what are the two critical points to establish?

You must show that the function is continuous and find points where it takes on negative and positive values.

How can the function f(x) = e^x - 3cos(x) + 1 be interpreted in terms of the Intermediate Value Theorem?

To show that it has at least one solution by demonstrating continuity and identifying values at specific points, one negative and one positive.

What is the definition of an average rate of change?

The ratio of the change in the function's value to the change in the variable over a specified interval.

What does the instantaneous rate of change refer to?

The limit of the average rate of change as the interval approaches zero.

What is meant by a function being differentiable at a point?

The derivative exists at that point, meaning the limit of the difference quotient is finite.

How can you determine the equation of a tangent line to a function at a specific point?

By finding the derivative at that point for the slope, then using the point-slope form of a line.

When might a function not be differentiable?

At points where there is a corner, vertical tangent, or the function is not continuous.

What is the geometric meaning of the derivative?

It represents the slope of the tangent line to the curve at a given point.

How do you find the equation of a line in vector form through two points in space?

Identify a direction vector by subtracting the coordinates of the two points and use one of the points as the position vector.

What constitutes a solution to the equation Ax = 0?

Any vector that when multiplied by the matrix A results in the zero vector.

What does it indicate if you have a free variable in a system of linear equations?

It suggests that there are infinitely many solutions to the system.

What is the implication of a matrix being in reduced row echelon form?

It easily reveals the relationships between the equations and allows for reading off solutions directly.

What does the null space of a matrix represent?

The set of all solutions to the equation Ax = 0, forming a vector space.

What is a key relationship between linear dependence and pivot columns in a matrix?

If a column does not contain a pivot, it can be expressed as a linear combination of the preceding columns.

How do matrix transformations relate to vector spaces?

Matrices can be viewed as linear transformations that map vectors from one vector space to another.

What is the definition of span in the context of vectors?

The span of a set of vectors is the set of all possible linear combinations of those vectors.

Under what condition is the system consistent if vector w is in the span of vectors u and v?

The system is consistent if there exist values for x1 and x2 such that x1 * u + x2 * v = w.

What value of h makes the vector w = (3, -5, h) an element of the span of u and v?

h must equal 3.

What are the characteristics of a consistent system in terms of linear combinations?

A system is consistent if the vector on the right-hand side can be expressed as a linear combination of the vectors on the left-hand side.

What does a pivot in every row of a coefficient matrix indicate about the span of its columns?

It indicates that the columns span the whole of R^m, meaning that the system Ax = b is consistent for every b.

What condition must be true for a vector b to be part of the span of vectors from a matrix?

Vector b must be expressible as a linear combination of the columns of the matrix.

When are two vectors linearly dependent?

Two vectors are linearly dependent if one is a scalar multiple of the other.

What happens if the number of vectors exceeds the number of dimensions in a space?

The set of vectors must be linearly dependent.

What does it mean if there is a free variable in a linear system?

It indicates that the system has infinitely many solutions, implying linear dependence of the columns.

How can you determine if a set of vectors is linearly independent?

A set of vectors is linearly independent if the equation Ax = 0 has only the trivial solution x = 0.

What must be true about a set of three vectors in R^3 for them to be coplanar?

At least one of the vectors must be expressible as a linear combination of the others.

What does a row of zeros in echelon form of a matrix imply about its columns?

It implies that not all columns have pivot positions, likely indicating linear dependence among the vectors.

What is meant by linear independence in relation to the trivial solution?

Linear independence means that the only solution to the vector equation is the trivial solution where all coefficients are zero.

What is the geometric interpretation of linearly dependent vectors in R^2?

Linearly dependent vectors in R^2 lie on the same line through the origin.

What does it mean for the span of a set of vectors to encompass the whole of R^m?

It means that every vector in R^m can be expressed as a linear combination of the vectors in the set.

What is the significance of echelon form in determining spans?

Echelon form helps identify pivot positions, which indicate whether a set spans R^m or if the system has inconsistent solutions.

What is a linear combination of vectors?

A linear combination of vectors is a weighted sum of those vectors, expressed as c1*v1 + c2*v2 + ... + cp*vp, where c1, c2, ..., cp are weights.

How is the span of a set of vectors defined?

The span of a set of vectors is the set of all possible linear combinations of those vectors.

What does it mean for a set of vectors to be linearly independent?

A set of vectors is linearly independent if the only linear combination that results in the zero vector is the trivial combination where all coefficients are zero.

What is the relationship between span and the entire R^2 plane?

The span of two non-collinear vectors in R^2 is the entire plane R^2, meaning any vector in R^2 can be expressed as a linear combination of them.

What is a defining characteristic of the zero vector in the context of span?

The zero vector is always included in the span of any set of vectors since you can obtain it by setting all weights to zero.

What is the outcome if a matrix A acts on a vector x in terms of linearity?

The action of a matrix A on a sum of vectors is the same as the sum of the actions of A on each vector, and multiplying a vector by a scalar before A yields the same result as multiplying the output by that scalar.

What type of geometric object does the span of two vectors in R^3 generate?

The span of two non-parallel vectors in R^3 generates a plane through the origin.

How does matrix-vector multiplication relate to linear combinations?

Matrix-vector multiplication can be viewed as a linear combination of the columns of the matrix, weighted by the entries of the vector.

What is the formula for matrix-vector multiplication?

If A is an m by n matrix and x is a vector in R^n, then Ax is defined as x1*A1 + x2*A2 + ... + xn*An, where Ai are the columns of A.

What signifies a consistent system in terms of an augmented matrix?

A consistent system has a solution, meaning there is no row in the row echelon form that leads to a contradiction such as 0 = non-zero number.

What are the three basic notions of calculus discussed in this lecture?

Limits, derivatives, and integrals.

What does the definition of limit represent in calculus?

It encompasses how a function behaves as it approaches a certain point, regardless of whether it is defined at that point.

For a function to have a limit L as x approaches a, what are the two conditions that need to be met?

1) The function must be defined in neighborhoods around a (excluding a itself). 2) For every positive number epsilon, there exists a positive number delta such that if x is within delta of a, then f(x) is within epsilon of L.

What does DNE stand for in the context of limits?

Does Not Exist.

What is implied when a function has an infinite limit?

The limit approaches infinity as x approaches a specific value, but this situation is treated differently from finite limits.

What is an interval in the context of limits?

A set of real numbers between two endpoints that defines the 'closeness' of the variable x to a point a.

What happens to the limit of the function f(x) = x as x approaches a?

The limit is equal to a, since f(x) outputs the same value as x.

Why does the function f(x) = sin(1/x) not have a limit as x approaches 0?

Because it oscillates infinitely between -1 and 1 without approaching any specific value.

In mathematical notation, what does the expression |f(x) - L| < epsilon signify?

It expresses that the value of the function f(x) is within epsilon distance from the limit L.

When discussing continuity, what does it mean for a function to be continuous at a point?

You can draw the function without lifting your pencil; the limit matches the function value at that point.

What graphical tool was mentioned in the lecture for visualizing functions?

Desmos.

In the context of limits, what does it mean if a function approaches different values from the left and right at a point?

The limit does not exist at that point.

When x approaches a certain value, what does it mean if the values of f(x) approach a different number?

The limit of the function could still be that different number even if the function value at that point is not defined.

What is the significance of epsilon and delta in the definition of a limit?

Epsilon represents how close f(x) needs to be to the limit, while delta represents how close x needs to be to the value a.

What is the intuition behind limits when two functions agree in a neighborhood, except maybe at a single point?

If functions f and g agree over a punctured interval around a point a, their limits as x approaches a are the same.

What is an indeterminate form in limits?

An indeterminate form occurs when both the numerator and denominator approach zero.

What is the result of the limit of the sum of two functions, provided their limits exist?

The limit of the sum is equal to the sum of the limits.

How do you define a left limit of a function f as x approaches a?

The left limit is the value that f approaches as x approaches a from values smaller than a.

What does it mean for a function to have a limit that approaches infinity?

It means that as x approaches a, the values of the function grow larger without bound.

In what case does the limit of a product exist?

The limit of a product exists if both functions involved have limits, and at least one of them is not zero.

How can you express the limit of f(x) as x approaches a if f is constant?

The limit of a constant function f(x) is simply the constant value itself.

What happens to the overall limit if the left limit and the right limit do not agree?

The overall limit does not exist.

Define the notation for a limit as x approaches a from the left.

The limit is written as lim (x → a-) f(x).

What effect does a vertical asymptote have on a function at a point a?

The function approaches infinity or negative infinity as x approaches a, but it does not actually touch or equal those values.

What is the Squeeze Theorem?

If g(x) is sandwiched between f(x) and h(x) and both f(x) and h(x) approach the same limit L as x approaches a, then g(x) must also approach L.

What is an indeterminate form involving limits?

When both the numerator and denominator approach infinity or zero, making the limit undefined without further analysis.

When x approaches a finite value and the function goes to infinity, what is this called?

An infinite limit.

What happens when you add a finite number to infinity?

The result is still infinity.

What is the key idea of limits at infinity?

To analyze the behavior of a function as x approaches positive or negative infinity.

What is the definition of continuity at a point?

A function f is continuous at a point a if the limit of f(x) as x approaches a exists and is equal to f(a).

What characterizes an 'infinite limit' at a point?

The function value tends toward positive or negative infinity as x approaches the point.

What does it mean for a function to oscillate infinitely without approaching a limit?

The function does not stabilize around any finite value as x approaches the point.

Give an example of a function that does not have a limit at a point due to oscillation.

The function sin(1/x) as x approaches 0.

What is the formula for the length (norm) of a vector in n-dimensional Euclidean space?

The length of vector v = (v1, v2,..., vn) is given by ||v|| = sqrt(v1^2 + v2^2 + ... + vn^2).

What are the standard unit vectors in R^2?

e1 = (1, 0) and e2 = (0, 1).

What does 'linear combination' of vectors mean?

A vector that can be formed by summing together multiples of those vectors.

What is an example of an indeterminate form?

0/0 or ∞ - ∞.

When can we apply the rules for limits for finite numbers only?

When both limits being added, subtracted, or multiplied are finite.

If two functions are bounded between g(x) and h(x) and both approach the limit L, what can we conclude?

g(x) must also approach L.