Calculus Flashcards
A curve is convex when..
f”(x) > 0
A curve is concave when…
f”(x) < 0
What is a stationary point of inflection?
Where f’(x) = 0 (stationary point) and where f”(x) = 0, where the gradient is positive on one side of this and negative on the other
After implicit differentiation you are usually left with a fraction such as (3x+2y) / (2x-6)
How do you then find the stationary point from this derivative?
Make the top of the fraction equal zero
So 3x + 2y = 0 —-> y = -3x/2
Substitute this is for any y in the original equation, then solve for x.
Don’t forget to get the y coordinate from the x values too!
Useful points for differentiating inverse trig functions (y = arcsin x)
First get rid of the arcsin by sin both sides. This will give you sin y = x which YOU WILL NEED TO SUBSTITUTE LATER ON
Use implicate differentiation to get dy/dx
Now substitute!
How do you integrate parametric equations?
- Differentiate x in terms of t
- Multiply dx/dt and y
Now you can integrate this!
If you’re integrating parametric equations, what must you remember to do?
Change the limits!
What is the general rule for integrating (1/ax+b)
1/a ln |ax+b| + C
If you need to integrate a fraction, what do you need to look out for?
See if the top is the derivative of the bottom (it may be multiplied by something too!)
If you need to integrate a fraction and there is no link between the top and bottom, what do you do now?!
You must split it up into partial fractions, then integrate as normal. Remember you may need to use the General Rule
What can you use to integrate functions such as cos^2 x, or sinxcosx?
Convert them using the double angle formulas, it will make it easier to integrate.
May need to rearrange to get one as the subject
If you’re finding it difficult to integrate a trig function (Such as cosec, sec, etc) , what can you do?
Rearrange it using the known identities and find a solution.
Remember to check your formula book for the given derivatives and integrals!
