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Material Type: Notes; Professor: Ku; Class: CALCULUS I; Subject: Mathematics; University: Montgomery College; Term: Unknown 1989;
Typology: Study notes
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h 13 ft
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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?
h
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?
Absolute Maximum:
Absolute Minimum:
Local Maximum:
Local Minimum:
Example 1 Example 2
Example 3 Example 4
Extreme Value Theorem
How do we find local extrema?
Fermat's Theorem:
Definition of a critical number:
Example 5
Fermat's Theorem Revised
Strategy for finding absolute extrema on a closed interval
Example 6
them on your graphing calculator.
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
2 / 3 1/ 3 y = x 6 − x
Example 1
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
Example 4 Graph the curve with parametric equations
in a viewing rectangle that displays all the important features of the curve and find the coordinates of all interesting points on the curve.
3 km C
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.
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.
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
x n , n ≠ − 1 sec^ x^ tan x
1 x^2
1 − x
e^ x cos x
2
1 + x
Example 5
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?