CECOR 1: Mathematics, Surveying and Transportation Engineering

Differential Calculus

Limits Slope of a Curve

The following standard methods (tricks) can be used The slope of the curve y=f(x) at any point is identical
to determine limits. to the derivative of the function or y’.
 If the limit is taken to infinity, all terms can be
divided by the largest power of x in the Rate of Change
expression. This will leave at least one constant. The derivative of a function is identical to its rate of
Any quantity divided by a power of x vanishes as change. Thus, the rate of change of the volume V of
x approaches infinity. a sphere with respect to its radius r is dV/dr.
 If the expression is a quotient of two
expressions, any common factors should be Curvature
eliminated from the numerator and denominator. 𝑦"
𝑘=
 L’Hopital’s should be used when the numerator (1 + 𝑦 ′2 )3/2
and denominator of the expression both
approach zero or both approach infinity. 𝑃𝑘 (𝑥) Radius of Curvature
and 𝑄 𝑘 (𝑥) are the kth derivatives of the (1 + 𝑦 ′2 )3/2
𝜌=
functions P(x) and Q(x) are the kth derivatives |𝑦"|
of the functions P(x) and Q(x), respectively.
(L’Hopital’s rule canbeapplied repeatedly as Center of Curvature
required.) 𝑦 ′ (1 + 𝑦 ′2 ) 1 + 𝑦 ′2
𝑎=𝑥− 𝑏=𝑦+
𝑦" 𝑦"
𝑃(𝑥) 𝑃𝑘 (𝑥) Complex Mode (Mode 2)
lim ( ) = lim ( 𝑘 )
𝑥→𝑎 𝑄(𝑥) 𝑥→𝑎 𝑄 (𝑥) (1 + 𝐴2 )3/2 −𝐴 + 𝑖
𝐶= :𝐷 +𝐶( )
𝐵 | −𝐴 + 𝑖 |
L’Hopital’s rule should not be used when only the Calc
A? y’
denominator approaches zero. In that case, the
B? y”
limit approaches infinity regardless of the D? x+yi
numerator.
Radius of Curvature (Polar Equation)
(𝑟 2 + 𝑟 ′2 )3/2
𝜌= 2
𝑟 + 2𝑟 ′2 − 𝑟𝑟"

Maxima and Minima (Critical Points)

Consider the graph of y=f(x). At the maximum and
minimum points, the slope is zero or the tangent is
horizontal.
𝑑𝑦
= 𝑦′ = 0
𝑑𝑥
These are the highest and lowest points relative to
the adjacent points.

Points of Inflection
Point of inflection is a point at which the curve
changes its rotation from concave upward to
concave downward or vice versa.
𝑦" = 0 𝑎𝑡 𝑝𝑜𝑖𝑛𝑡𝑠 𝑜𝑓 𝑖𝑛𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛

MAAM | Rev0 Page 1 of 2

CECOR 1: Mathematics, Surveying and Transportation Engineering
Differential Calculus

Problems: 14) A man on an island 12 km sought of a straight

Situation No. 1 beach wishes to reach a point on shore 20 km
Evaluate the following limits west. If a motorboat, making 20 kph, can be
𝑥 3 −27 hired at the rate of P1000 per hour for the time it
1) lim ( 𝑥 2 −9 ) is actually used, and the cost of land
𝑥→3
3𝑥−2 transportation is P30 per kilometer, how much
2) lim (4𝑥+3) must he pay for the trip?
𝑥→∞
𝑥 2 +𝑥−6 15) A 5-meter line AD is perpendicular to another 5-
3) lim (𝑥 2 −3𝑥+2) meter line BC. The two lines intersect at point D
𝑥→2
4) Find the angle of intersection of the following so that BD=2 meters and DC=3 meter. Point P
pairs of curve xy+y=1 and y^3=(x+1)^2. on line AD is x meters away from A. The total
5) Find the slope of the curve 𝑟 2 = cos 2𝜃 at 𝜃 = length of the lines linking P to point A, B, and C
𝜋/6. is minimized. Find x.
6) Find the point of inflection of the curve 𝑦 = 𝑥 3 − 16) A rectangle box with square base and open at
3𝑥 2 − 𝑥 + 7. the top is to have a capacity of 16823 cu.m. Find
7) Find the curvature of the parabola x=y^3 at (2,1). the height of the box that requires minimum
8) An inverted conical tank has a base radius of 10 amount of material required.
m and height of 20 m. Water flows out from the 17) Find the sum of two numbers whose sum is 36 if
tank at 150 liters per second. How fast is the the product of one by the square of other is a
water surface falling when the depth of water in maximum.
the tank is 8 m?
9) Gas escaping from a spherical balloon at the rate
of 2 cc/min. Find the rate at which the surface
area is decreasing, in cm^2/min, when the radius
is 8 cm.
10) A Ferris wheel with radius of 10 m has its center
12 m above the ground. The seats of the wheel
have vertical speeds of 1.81 m/s when it is 17 m
above the ground. How fast is the wheel rotating
in revolutions per minute?
11) A bomber plane, flying horizontally 3.2 km above
the ground is sighting on at a target on the
ground directly ahead. The angle between the
line of sight and the pad of the plane is changing
at the rate of 5/12 rad/min. When the angle is 30
degrees, what is the speed of the plane in mph?
12) Suppose that x years after founding in 1975, a
certain employee association had a membership
of f(x)=100(2x^3-45x^2+264x), at what time
between 1975 and 1989 was the membership
smallest?
13) Determine the shortest distance from point (4, 2)
to the parabola y^2=8x

MAAM | Rev0 Page 2 of 2