Core Modus Flashcards
f(x) is continuous at x=a if?
f(a) exists
lim(x>a) f(x) exists
lim(x>a) f(x) = f(a)
x intercept is where?
y=0
and vice versa
means make y=0 in equation from standard form
equation of a given line?
y-y1 = m(x-x1)
where x1 & y1 is a point on the line
when given two points
(x1, y1) (x2, y2)
rise over run to find gradient
m = y2-y1/x2-x1
perpendicular lines
meet at right angles
m1*m2 = -1 always
parallel lines
always have the same gradient
to get m (gradient) you must
convert to gradient intercept form
quadratic function
form?
y = ax2+bx+c
line of symmetry formula
x = -b/2a
from ax2+bx+c
find Turning Point
substitute the line of symmetry into x of
y = ax2+bx+c
if Line of Symmetry x = -1 and evaluating gives y = -4
then Turning Point is (-1, -4)
cubic function polynomial form
f(x) = ax3+bx2+cx+d
quartic function polynomial form
f(x) = ax4+bx3+cx2+dx+e
3/5 / 5
3/5 / 5
3/25
3/5 * 5
3/5 * 5
3
-x/3
also looks like
- 1/3 x
y = -3x+4
make y negative
-y = 3x-4
inverting all values keeps the balance
find inverse function
template?
- sub y for f(x)
- make x subject
- switch x and y
- write f-1(x) instead of y
ln(4)-3/2 =
ln(4)-3/2 =
1/2(ln(4)-3)
inverse function of y = ex
inverse function of y = ex
y = lnx
find Period
2π/B
where B is the coefficient of x
example: y = cos2x
B = 2
definition of the Derivative
dy/dx
= lim(h>0) f(x+h)-f(x)/h
rise/run

d/dx C
= 0
the derivative of any constant even when negative are equal to 0.
No rise, only run.
dy/dx of y = 3x
dy/dx of y = 3x
dy/dx
= d/dx 3x
= 3
Always equal to the coefficient/ gradient
table of derivatives
c <em>(a constant)</em> 0
axn naxn-1
sinx <em>(x in radians)</em> cosx
cosx (x in radians) -sinx
eax (a is constant) aeax
lnx or logex 1/x or x-1
dy/dx 16/x
dy/dx 16/x
= 16x-1
= (-1)16x-2
= -16x-2
= -16/x2
dy/dx 5√x
dy/dx 5√x
= 5x1/2
= 1/2(5)x- 1/2
= _5/2_x- 1/2
= 5/2√x
dy/dx 7e4x
dy/dx 7e4x
= 7 d/dx e4x
= 7 * 4e4x
= 28e4x
dy/dx 2lnx
dy/dx 2lnx
= d/dx 2lnx
= 2 d/dx lnx
= 2 * 1/x
= 2/1 * 1/x
= 2/x
dy/dx 3cosx
dy/dx 3cosx
= 3 d/dx cosx
= -3sinx
dy/dx -√2 sinx
dy/dx -√2 sinx
= -√2 d/dx sinx
= -√2 cosx
(6+x)(2-x)
x = ?
(6+x)(2-x)
x = -6 or 2
similar to horizontal shift
y = -x2
parabola shape?
negative frowney face
graph of a function vs its derivative
-y = 12-4x-x2
function
- y = 12-4x-x2 x-int @ y = 0
- 0 = 12-4x-x2 = (6+x)(2-x)
= -6 or 2
derivative
dy/dx = -4 -2x x-int @ dy/dx = 0
0 = -4 -2x
4 = -2x
4/-2 = x = -2

turning points occur when
dy/dx = 0