Chain, Chain, Chain
The derivative of
f(x) = sin (2x)
2cos(2x)
The derivative of
f(t) = sin 2t
2cos 2t
Sir Isaac Newton developed calculus in order to see these
Planets in the sky
int1/xdx
lnx+c
The equation of the position of a particle over time is modeled by s(t) = 3t^2 .
What's equation may represent its velocity?
v(t)=6t
The derivative of
f(x) = e^(2x^2)
4xe^(2x^2)
The derivative of
f(t) = - cos 2t
2sin 2t
Leibniz developed calculus so he could measure this
Area under a curve
int(sqrtx) dx
(2/3)x^(3/2)+c
The equation of the velocity of a particle over time is modeled by v(t) = 5t^2 - 2t
What equation may represent its acceleration?
a(t)=10t-2
The derivative of
f(x) = pi^(3x)
3pi^(3x)lnpi 3pi^(3x)ln(pi)
The derivative of
f(t) = -1/2tan t^2
-tsec^2(t^2)
The two calculus processes, differentiation and integration
Inverses
int3cos(3x) dx
sin(3x)+c
The equation of the acceleration of a particle with respect to time is modeled by a(t) = 4t^2 - 7t .
What equation may represent its velocity?
a(t) = 4/3t^3-7/2t^2+c
The derivative of
f(x) = ln (3x-x^2)
(3-2x)/(3x-x^2
The derivative of
f(t) = cos^2 (t)
-2cos(t)sin(t)
The derivative of any parabola
A straight line
intpix^2+7x dx
pi/3x^3+7/2x^2+c
The position of a particle with respect to time is represented by the equation s(t) = 3t- t^3 .
What equation may represent its acceleration?
a(t)=-6t=s''(t)
The derivative of
f(x) = sqrt(ln x)
1/(2xsqrtlnx)
The derivative of
f(x) = sqrt sec(t)
(tan tsect)/(2sqrtsect) or 1/2sin tsec^2t
A point of a curve at which a change in the direction of the curvature occurs.
Point of Inflection
int-csc^2(5x)dx
1/5cot(5x)+ C
The acceleration of a particle is represented by the equation a(t)=t^2-4t .
What equation may represent its position with respect to time?
1/3t^4/4-2/3t^3+ct+c
or
(t^4)/12-(2t^3)/3+c(t + 1)