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Differential CalculusIn the following formulas, , , and are
differentiable functions of and and are
constants.
Differentiation of Algebraic Functions
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Differentiation of Logarithmic and Exponential Functions
1.
2.
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3.
4.
5.
6.
Differentiation of Trigonometric Functions
1.
2.
3.
4.
5.
6.
Differentiation of Inverse Trigonometric Functions
1.
2.
3.
4.
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5.
6.
Differentiation of Hyperbolic Functions
1.
2.
3.
4.
5.
6.
Differentiation of Inverse Hyperbolic Functions
1.
2.
3.
4.
5.
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5.
Maxima and Minima | ApplicationsTags:maxima and minimapoint of
inflectioncritical pointcritical valueslopefirst derivativesecond
derivativerate of change
Graph of the Function y = f(x)
The graph of a function y = f(x) may be plotted using
Differential Calculus. Consider the graph shownbelow.
As x increases, the curve rises if the slope is positive, as of
arc AB; it falls if the slope is negative, as
of arc BC.
Relative Maximum and Minimum Points
At a point such as B, where the function is algebraically
greater than that of any neighboring point,
the point is said to have a maximum value, and the point is
called a maximum point (relative toadjacent points). Similarly at
D, the function has a minimum value (relative to adjacent points).
At
maximum or minimum points, the tangent is horizontal or the
slope is zero.
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This does not necessarily mean that at these points the function
is maximum or minimum. It does
only mean that the tangent is parallel to the x-axis, or the
curve is eitherconcave up orconcave
down. The points at which dy/dx = 0 are called critical points,
and the corresponding values of x
are critical values.
The second derivative of a function is the rate of change of the
first derivative or the rate of changeof the slope. It follows that
as x increases and y" is positive, y' is increasing and the tangent
turns in
a counterclockwise direction and the curve is concave upward.
When y" is negative, y' decreases
and the tangent turns in the clockwise direction and the curve
is concave downward.
If y' = 0 and y" is negative (i.e. y" < 0), the point is a
maximum point (concave downward).
If y' = 0 and y" is positive (i.e. y" > 0), the point is a
minimum point (concave upward).
Points of Inflection
A point of inflection is a point at which the curve changes from
concave upward to concavedownward or vice versa (see point E from
the figure). At these points the tangent changes its
rotation from clockwise to counterclockwise or vice versa.
At points of inflection, the second derivative of y is zero (y"
= 0).
pplication of Maxima and MinimaTags:maxima and minima
As an example, the area of a rectangular lot, expressed in terms
of its length and width, may also beexpressed in terms of the cost
of fencing. Thus the area can be expressed as A = f(x). The
common
task here is to find the value of x that will give a maximum
value of A. To find this value, we set
dA/dx = 0.
Steps in Solving Maxima and Minima Problems
1. Identify the constant, say cost of fencing.
2. Identify the variable to be maximized or minimized, say area
A.
3. Express this variable in terms of the other relevant
variable(s), say A = f(x, y).
4. If the function shall consist of more than one variable,
expressed it in terms of one variable (if
possible and practical) using the conditions in the problem, say
A = f(x).5. Differentiate and equate to zero, dA/dx = 0.
Time Rates | ApplicationsTags:time
ratesvelocityaccelerationflow
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dischargeangular speed
Time Rates
If a quantity x is a function of time t, the time rate of change
of x is given by dx/dt.
When two or more quantities, all functions of t, are related by
an equation, the relation between their
rates of change may be obtained by differentiating both sides of
the equation with respect to t.
Basic Time Rates
Velocity, , where is the distance.
Acceleration, , where is velocity and is the distance.
Discharge, , where is the volume at any time.
Angular Speed, , where is the angle at any time.
Steps in Solving Time Rates Problem
1. Identify what are changing and what are fixed.
2. Assign variables to those that are changing and appropriate
value (constant) to those that
are fixed.
3. Create an equation relating all the variables and constants
in Step 2.
4. Differentiate the equation with respect to time.
5. Chapter 4 - Trigonometric and Inverse
TrigonometricFunctions
6. Differentiation of Trigonometric Functions
Trigonometric identities and formulasare basic requirements for
this section. Ifu is a function
ofx, then
7. 1.
8. 2.
9. 3.
10. 4.
11. 5.
12. 6.
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13. Differentiation of Inverse Trigonometric Functions
In the formula below, u is any function ofx.
14. 1.
15. 2.
16. 3.
17. 4.
18. 5.
19. 6.
