Revision of Calculus and Elementary Curve Sketching

Question 1

Define derivative

Answer 1

dy/dx is the slope of the (local) tangent to y(x) at a point

Question 2

What are the conditions for a function to be differentiable at x=x(0)?

Answer 2

Differentiable at x=x(0) if:
lim as δx->0 for (f(x(0) + δx) - f(x(0)))/δx
must exist and not depend on the sign of δx

Question 3

How can a function be continuous at x=x(0)?

Answer 3

Continuous at x=x(0) if:

lim as x->x(0) f(x) exists and equals f(x(0))

Question 4

Define e^x by the power series

Answer 4

e^x = 1 + x + 1/2! x^2 + 1/3! x^3 + …

Question 5

What are the two log rules?

Answer 5

ln(xy) = ln(x) + ln(y)
ln(x^y) = yln(x)

Question 6

What do the hyperbolic functions differentiate to?

Answer 6

cosh(x) -> sinh(x)
sinh(x) -> cosh(x)
tanh(x) -> 1/cosh^2 (x)

Question 7

What are the four rules of differentiation?

Answer 7

Product rule: d(uv)/dx = u’v + uv’
Quotient rule d(u/v) = (vu’ - uv’)/(v^2)1
Chain rule: y(x) = f(g(x)), dy/dx = df/dg dg/dx = f’(g(x))g’(x)
Reciprocal rule: dx/dy = 1/(dy/dx)

Question 8

How would implicit differentiation work for y^3?

Answer 8

d(y^3)/dx = d(y^3)/dy dy/dx = 3y^2 dy/dx

Question 9

The general form of Leibnitz’s formula

Answer 9

d^n (fg)/dx^n = Sum (between n and m=0) of nCm f^(n-m) g^m

=f^(n)g^(0) + nf^(n-1) g^(1) + … + f^(0) g^(n)

Question 10

Pascal’s rule

Answer 10

nCm + nC(m+1) = (n+1)C(m+1)

Question 11

Define a maximum point

Answer 11

dy/dx = 0 and d^2 y/dx^2 < 0

Question 12

Define a minimum point

Answer 12

dy/dx = 0 and d^2 y/dx^2 > 0

Question 13

What is an inflection point?

Answer 13

The point at which d^2 y/dx^2 changes sign

Question 14

Things to think when drawing a graph

Answer 14

Symmetry
x and y-intercepts
Limiting behaviour for small and large x
Stationary points
Singular points and vertical asymptotes

Question 15

As x tends to infinity, order these: e^x, ln(x) and x^n in terms of which approaches ∞ the fastest

Answer 15

e^x, x^n, ln(x)

where n is constant and >0

Question 16

What is proportional to the width of distribution for Normal distribution?

Answer 16

1/a from e^(-ax^2)

Question 17

What is sinc(x)?

Answer 17

sinc(x) = sin(x)/x

Question 18

L’Hopital’s rule

Answer 18

Provided that lim as x->x(0) (f’/g’) exists

lim as x->x(0) f(x)/g(x) = lim as x->x(0) (f’/g’)

Question 19

If lim as x->x(0) f(x) = A and g(x) = B, what arises from this through addition, multiplication and division?

Answer 19

Addition: lim as x->x(0) [f(x) + g(x)] = A + B
Multiplication: lim as x->x(0) [f(x)g(x)] = AB
Division: lim as x->x(0) [f(x)/g(x)] = A/B for B =/= 0
lim as x->x(0) [f(x)/g(x)] doesn’t exist for A=/= 0 and B = 0
lim as x->x(0) [f(x)/g(x)] may exist for A=B=0
lim as x->x(0) [f(x)/g(x)] may exist for A=B=infinity

Question 20

What is the condition for absolute convergence?

Answer 20

Sum between ∞ and k=0 for |u(k)| converges

u(k) can be complex

Question 21

What is the condition for conditional convergence?

Answer 21

If the sum between ∞ and k=0 for u(k) converges, but the sum between ∞ and k=0 for |u(k)| does not converge

Question 22

What is the sum between n and k=0 for r^k equal to?

Answer 22

(1-r^(n+1))/(1-r)

Question 23

What is the sum between ∞ and k=0 for r^k equal to?

Answer 23

1/(1-r) where |r|<1

Question 24

What is the harmonic series?

Answer 24

Sum between infinity and k=1 for 1/k

Series diverges

Question 25

How can the comparison test be used to show a series converges?

Answer 25

If u(k) =< v(k) for all k >= some K, then sum between ∞ and k=0 for u(k) converges if the sum between ∞ and k=0 for v(k) also converges

Question 26

How can the comparison test be used to show a series converges?

Answer 26

If u(k) >= v(k) for all k > some K, then sum between ∞ and k=0 for u(k) diverges if the sum between ∞ and k=0 for v(k) also diverges

Question 27

How does the ratio test work?

Answer 27

Given a positive series sum of u(k) if:
lim as k->∞ u(k+1)/u(k) < 1 then sum of u(k) converges
lim as k->∞ u(k+1)/u(k) > 1 then sum of u(k) diverges
lim as k->∞ u(k+1)/u(k) = 1 then sum of u(k) may converge

Question 28

What is the Leibnitz criterion?

Answer 28

If the series alternates and terms monotonically decrease in magnitude as n increases, series converges

Question 29

What is the form of a power series?

Answer 29

f(x) = sum between ∞ and k=0 for a(k)x^k = a(0) + a(1)x + a(2)x^2 + …
defines the function f(x) provided series converges

Question 30

The general form of a Taylor series

Answer 30

f(x) = f(0) + xf’(0) + (x^2)/2! f’‘(0) + (x^3)/3! f’’‘(0) + …

Question 31

The general form of a Maclaurin series

Answer 31

f(x) = f(a) + (x-a)f’(a) + ((x-a)^2)/2! f’‘(a) + …

Question 32

What does ln(1+x) equal?

Answer 32

ln(1+x) = x - (x^2)/2 + (x^3)/3 - (x^4)/4 + …

Question 33

Newton-Raphson root-finding method

Answer 33

x(1) = x(0) - f(x(0))/f’(x(0))