Unformatted text preview: Summary of Calculus
Review of Dierentiation
Denition of the Derivative
f 0 (x) = lim h→0 f (x + h) − f (x)
h Properties of Dierentiation
d
d
d
[f (x) ± g(x)] =
[f (x)] ±
[g(x)]
dx
dx
dx
d
d
[k · f (x)] = k ·
[f (x)]
dx
dx
d
[k] = 0
dx Power Rule
Product Rule
Quotient Rule
Chain Rule d n
[x ] = n · xn−1
dx
d
[f · g] = f 0 g + g 0 f
dx
d f
f 0g − g0f
=
dx g
g2
dy
dy du
=
·
dx
du dx 1 Exponential and Logarithmic Table
Function Derivative
dy
dx y = ex dy
dx y = ax = ax ln a dy
dx y = ln x dy
dx y = loga x 2 = ex = = 1
x 1
x ln a Trig Table
Function Derivative
dy
dx y = sin x dy
dx y = cos x dy
dx y = sec x y = csc x = − sin x dy
dx y = tan x dy
dx y = cot x = cos x = sec2 x = sec x tan x = − csc x cot x dy
dx = − csc2 x Inverse Trig Table
function derivative y = sin−1 x y0 = y = cos−1 x √ 1
1−x2 1
y 0 = − √1−x
2 y = tan−1 x y0 = 3 1
1+x2 Review of Calculus  Applications of Dierentiation
Rates of Change
s(t) =
v(t) = s0 (t) =
a(t) = v 0 (t) = s00 (t) = distance function
velocity function
acceleration function Optimization (Maximization and Minimization)
Obtain a function
equate to f (x) in terms of one variable, then obtain f 0 (x), and lastly 0. Gradient Function and Tangent Lines
f (x) = function
f 0 (x) = gradient function
Finding Relative and Absolute Extrema  First Derivative
Test (FDT)
To nd the critical points and to classify them as relative maximum or relative minimum, we use the First Derivative Test
Steps:
1. 2. Find all the critical points of f (x) by equating f 0 (x) = 0 and
solve for x.
Then use the Sign Test using f 0 (x) to determine the nature
of the critical point x0 .
(a) (b) (c) If the values of f 0 (x) are positive on the left of x0 and
negative on the right of x0 , then x0 is a relative maximum.
If the values of f 0 (x) are negative on the left of x0 and
positive on the right of x0 , then x0 is a relative minimum.
If the values of f 0 (x) are positive (or negative, resp.)
on the left of x0 and positive (or negative, resp.) on
the right of x0 , then x0 is not a relative extrema.
4 Point of Inection and Concavity
Given a function f (x),
1. 2. dierentiate f (x) twice [f 00 (x)] and equating it to 0 to nd
point of inection x0 .
Use Sign Test to nd concavity:
(a) (b) For the region of positive f 00 (x) values, we have concave up
(cup facing up).
For the region of negative f 00 (x) values, we have concave
down (cup facing down). 5 Review of Integration
The Indenite Integral (Antiderivative)
ˆ F (x) = f (x)dx The Denite Integral (Area under the Curve)
ˆ b f (x)dx A=
a Integration Table for Polynomials
function Integration formula
´ 1
´ x ´ x2 ´ x3 xn , n 6= −1 ´ 1dx = x + c xdx = 12 x2 + c
x2 dx = 31 x3 + c
x3 dx = 41 x4 + c xn dx = 6 1
xn+1
n+1 +c Integration Table for Trig, Exp, Log
function integration − sin x −
´ sin x ´ sec2 x csc2 x 1
x ex ´ sin xdx = cos x + c sin xdx = − cos x + c
´ cos x csc x cot x ´ cos xdx = sin x + c sec2 x = tan x + c csc x cot xdx = − csc x + c
´ ´ csc2 xdx = − cot x + c 1
dx
x ´ = ´ dx
x = ln x + c ex dx = ex + c 7 Integration by usubstitution
For ˆ
f (g(x)) · g 0 (x)dx, let u = g(x), then du = g 0 (x)dx and
ˆ ˆ 0 f (g(x)) · g (x)dx = Integration by Parts ˆ f (u)du. ˆ
u dv = uv − 8 v du. Review of Numerical Integration
Midpoint Approximation
For Midpoint approximation, we have the following formulae to use:
width of the subintervals, ∆x choice of midpoints of the subintervals, ˆ b−a
n = 1
x∗k = a + (k − )∆x
2 b f (x)dx ≈ Mn = [f (x∗1 ) + f (x∗2 ) + . . . + f (x∗n )] · ∆x Area function,
a Trapezoidal Approximation
Also known as Trapeziodal Rule. The formula to use are:
width of the subintervals, ˆ b f (x)dx ≈ Tn = Area function
a ∆x = b−a
n 1
[y0 + 2y1 + 2y2 + . . . + 2yn−1 + yn ] · ∆x
2 Simpsons Approximation
Also known as Simpsons Rule. The formula to use is: ˆ b f (x)dx ≈ S2n =
a 1
3 b−a
2n
y0 + 4y1 + 2y2 + 4y3 + 2y4 + . . . + 4y(2n)−1 + y2n . 9 ...
View
Full Document
 Spring '16
 Alveen Chand
 Calculus, Derivative, dx, Di1Berentiation

Click to edit the document details