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4.1 Related Rates
The idea: If we can find a formula relating two quantities then by taking the derivative of both sides (with respect to some third quantity, usually time) we have a formula relating their derivatives (rates).
Example 1 Air is being pumped in to a balloon so that its volume is increasing at a rate of 100 cm^3 /s. How fast is the radius of the balloon increasing when the diameter is 50 cm?
Example 2 A 13 ft ladder is resting against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1 ft/s, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 5 ft from the wall?
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Example A water tank has the shape of an inverted circular cone with base radius 2 m and height 4 m. If water is being pumped in to the tank at the rate of 2 m^3 /min, find the rate at which the water level is rising when the water is 3 m deep.
Example 4 Car A is travelling west at 50 mi/hr and car B is travelling north at 60 mi/hr. Both are headed for the intersection of the two roads. At what rate are the cars approaching each other when car A is 0.3 mi and car B is 0.4 mi from the intersection?
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4.2 Maximum and Minimum Values
Absolute Maximum:
Absolute Minimum:
Local Maximum:
Local Minimum:
Example 1 Example 2
The function f ( x ) = cos( x ) The function f ( x ) = x^2
Example 3 Example 4
The function f ( x ) = x^3 The function f ( x ) = 3 x^4 − 16 x^3^ + 18 x^2 , − 1 ≤ x ≤ 4
Extreme Value Theorem
How do we find local extrema?
Fermat's Theorem:
Definition of a critical number:
Example 5
Find the critical numbers for f ( x ) = x 2 / 3^ ( 2 + x )
Fermat's Theorem Revised
Strategy for finding absolute extrema on a closed interval
Example 6
Find the exact maximum and minimum values of f ( x ) = x − 2sin x over the interval 0 ≤ x ≤ 2 π and check
them on your graphing calculator.
HW 4.2 # 3, 5, 7, 11, 13, 15, 17, 21, 23, 29, 31, 37, 39, 43, 45, 53
The First Derivative Test
Find the local maximum and minimum values of the function from the previous example.
Concavity
Concavity Test
Second Derivative Test
Example 2 Use the second derivative test on the previous example
Example 3
Figure out everything you can about y = x^4 − 4 x^3 with respect to concavity, points of inflection, and local
maxima and minima. Use this information to sketch a graph.
Example 4
Sketch the graph of ( )
2 / 3 1/ 3 y = x 6 − x
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4.4 Graphing with Calculus and Calculators
Example 1
Graph the polynomial f ( x ) = 2 x^6^ + 3 x^5^ + 3 x^3 − 2 x^2. Use the graphs of f ′^ ( x )and f ′′^ ( x )to estimate all
maximum and minimum points and intervals of concavity.
Example 2 Draw the graph of the function
2 2
x 7 x 3 f x x
Find the maximum and minimum values, and the intervals of concavity.
Example 3
Graph the function f ( x ) = sin ( x + sin 2( x ))and estimate all extrema and intervals of concavity.
Example 4 Graph the curve with parametric equations
x ( ) t = t^2 + t + 1 y t ( ) = 3 t^4 − 8 t 3^ − 18 t^2 + 25
in a viewing rectangle that displays all the important features of the curve and find the coordinates of all interesting points on the curve.
HW 4.4 # 1, 7, 9, 11, 21
A
B
3 km C
D
8 km
Example 4
Find the point on the parabola 2
y = x that is closest to the point (4,1)
Example 5 A man launches a boat from point A on a bank of a straight river 3 km wide and wants to reach a point B, 8 km downstream on the opposite bank as quickly as possible. He could row directly to B, or he could row directly across the river to point C and run to B, or he could row to some point D between B and C and then run to B. If he can row at 6 km/h and run at 8 km/h, where should he land to reach B as soon as possible? (We assume that the speed of the water is negligible compared with the speed at which the man rows)
Example 6 Find the area of the largest rectangle that can be inscribed in a circle of radius r.
HW 4.6 # 5, 9, 11, 13, 19, 21, 25, 37, 39
Example 2
Use Newton's Method to estimate 6 2 to 8 decimal places.
Example 3 Estimate the solution to cos x = x to 6 decimal places.
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4.9 Antiderivatives
Definition of an antiderivative :
Theorem :
Example 1 Find the general antiderivative of each of the following
a) cos x b) 2 x c) x n , n ≠ − 1 d)
x
Antidifferentiation Table
We let F ′ = f and G ′ = g
Function Particular antiderivative Function Particular Antiderivative
c f ( x ) sin x
f ( x ) + g ( x ) sec^2 x
x n , n ≠ − 1 sec^ x^ tan x
1 x^2
1 − x
e^ x cos x
2
1 + x
Example 5
A particle moves in a straight line and has acceleration given by a t ( ) = 6 t + 4. Its initial velocity is
v ( 0 )= − 6 and its initial displacement is s ( 0 )= 9. Find its position function s t ( ).
Example 6 A ball is thrown upward with a speed of 48 ft/s from the edge of a cliff 432 ft above the ground. Find its height above the ground t seconds later. When does it reach its maximum height, when does it hit the ground?
HW 4.9 # 1, 3, 7, 13, 15, 17, 25, 39, 45
