Calc

Question 1

P(x)/Q(x)
Rational functions have asymptotes. An asymptote is ____. A rational function P(x)/Q(x) has vertical asymptotes only where ____
A horizontal asymptote only exists if degree(P)____ (≥/≤)degree(Q).

P16

Answer 1

A straight line which is approached by the graph of the function
The denominator equals 0, so when Q(x) = 0.
≤
It can be calculated by dividing the numerator and denominator by the highest power of x occurring in the denominator, and noting that 1/xn goes to 0 as x becomes very large.

The function f(x) = 1/x is a rational function. It has vertical asymptote x = 0.
The horizontal asymptote is y = 0 as 1/x goes to 0 when x becomes large.

Question 2

The function f(x) = e^x
is special because the growth rate
of this function is directly proportional to the value of the function (or in other words:____)

P17

Answer 2

the function’s derivative is equal to itself.

Question 3

tan’s period is ____
sin(2x) = ____
cos(2x) = ____
sin(x + y) = ____
cos(x + y) = ____

P19

Answer 3

π
2 sin(x) cos(x)
cos²(x) − sin²(x)
cos(x) sin(y) + sin(x) cos(y)
cos(x) cos(y) − sin(x) sin(y)

Question 4

• If x = α is a solution to sin(x) = c, then x = ____ and x = ____ are also solutions for all integers k.
• If x = β is a solution to cos(x) = c, then x = ____ and x = ____ are also solutions for all integers k.
• If x = γ is a solution to tan(x) = c, then x = ____ are also solutions for all integers
  k.

P20

Answer 4

α + 2kπ
π − α + 2kπ (because sin(π − x) = sin(x))

β + 2kπ
−β + 2kπ

γ + kπ

Question 5

Derivatives to know

P26

Question 6

How is f(x)/g(x) differentiated

P26

Answer 6

Answer

Question 7

A point x for which f’(x) = 0 is called a stationary point of a differentiable function. To see if a stationary point is a local minimum, a local maximum, or neither(saddle point like in X³ at 0), you have to use ____ test or create a ____.

P28

Answer 7

the second derivative: Given a function f(x) that is twice differentiable at a point x0, where x0 is a point
such that f’(x0) = 0.
* If f’‘(x0) > 0 then f has a local minimum at x0.
* If f’‘(x0) < 0 then f has a local maximum at x0.

sign chart: A sign chart is a chart that shows the behaviour of f between stationary points, i.e. is the function increasing or decreasing. For example, if a function is increasing before a stationary point and decreasing after, we know we deal with a local maximum

Remember: After we’ve found points where the function changes its direction in sign chart, we must check whether the function is defined at those points. Otherwise, this isn’t a relative extremum.

Question 8

Important Integrals

P32

Question 9

Property of Integrals

P33

Question 10

Substitution rule

P34

Question 11

Integration by parts
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