Calculus - Calculus (5) Flashcards
Finding the co-ords of a tangent when given equations
- Get the two derivatives
- Set them equal, find x(s)
- Put x(s) into tan equation
- Find y by original equation
- State co-ordinate
Drawing the gradient of a cubic
- Local min/max as the x-ints
- Maximum point in the middle of the points
Maximum area with given fencing, etc
- 2x+2y=perimeter. xy=area
- Find x in terms of y, replace x in the equations with y.
- Differentiate & set to zero (finding the max)
- Find y, then use og equation to find x.
- State the max values & area.
Anti-differentiate to find the original equation with given co-ords
- Anti-differentiate + c
- Plug in x&y. Rearrange for c
- Add c into the eq we found
Find the equation of a tangent, given the co-ords & the og equation
- Differentiate
- Find the gradient using the x value
- Use y-y1=m(x-x1) to find the equation
Drawing the original curve from the differentiated one
- Use the x-intercept as the maximum value
- Make sure it’s going the right way
Find ‘a’, etc, using the gradient & equation
- Differentiate
- Set gradient equal to …
- Solve for a
Find the distance from the velocity equation
- Anti differentiate to get the distance equation
- Find C if relevant
- Solve equation
Kinematics process
s(t) s’(t) s’‘(t)
v(t) v’(t)
a(t)
>Differentiate
<Anti-differentiate
Two things travelling at different speeds, how long until they meet?
- Get the velocity equations. For the one behind, minus the distance
- Set them equal
- Make it equal 0 if ^2 is involved
- Solve for time
Equation with unknowns, how to find a & b. Given co-ords.
- Differentiate, set to zero.
- Find a in terms of b, or vice versa
- Sub in a in the 1st eq for new formula for b
- Put in the co-ords to find a.
- State what a equals, use to find b.
- Sub these into the original equation, differentiate, set it to 0 to find the other turning point
- Use x to find y, state the co-ordinates
- Differentiate again to see if local min/max
When x^3 is in an equation
Factorize everything, make make x=0 is one of the answers (it has 3 bumps)
We have the original equation and the tangent function, we need to prove this
- Use y=mx+c to find the m.
- Differenciate the og and plug in that m to find x.
- Use the og equation to find y. Might have two points.
- Y-y1=m(x-x1) for both
When the gradient is maximum, when something is growing fastest
Differentiate twice. Can then plug in that the gradient = 0 and x value.
When 2x(x-4)
X = 0 and X=4! Always has two answers
