Lesson 3.2 Rolle and Mean Value Theorems
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Rolle's Theorem states that if a function f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), with f(a) = f(b), then there exists at least one value c in the open interval where the derivative f'(c) is equal to 0. The Mean Value Theorem similarly guarantees the existence of at least one value c where the derivative is equal to the average rate of change of the function over the interval, (f(b)-f(a))/(b-a).
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![Rolle’s Theorem
1. If f is continuous on [a,b], AND
2. Differentiable on (a,b), AND
3. f(a) = f(b), THEN…
…you are guaranteed at least one value c on (a,b) where f’(c) = 0.
a
b](https://image.slidesharecdn.com/lesson3-131109074339-phpapp01/85/Lesson-3-2-Rolle-and-Mean-Value-Theorems-1-320.jpg)
![Mean Value Theorem
1. If f is continuous on [a,b] AND
2. Differentiable on (a,b), THEN…
f ( b) − f ( a )
…there is at least one value c on (a,b) where f ′( c ) =
b−a
a
b](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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This document defines and provides examples of related rate problems. Related rate problems involve calculating the rates of change of two or more related variables with respect to time. The document outlines the steps to solve related rate problems: 1) identify changing quantities, 2) write an equation relating the quantities, 3) differentiate both sides with respect to time, 4) solve for the required quantity and substitute known values. Examples include calculating the rate of change of a dog's circular puddle area, a melting snowball's radius, the top of a slipping ladder, and the height of a conical sand pile.
3.5 EXP-LOG MODELS
The document discusses various mathematical models involving exponential and logarithmic functions, including exponential growth and decay models, logistic growth models, Gaussian models, and logarithmic models. It provides examples of how each type of model can be used, such as modeling world population growth with an exponential function, modeling the spread of a virus on a college campus with a logistic function, and modeling cooling of a heated object using Newton's Law of Cooling. The document also discusses how to graph and solve problems involving these different mathematical models.
Larson 4.4
This document discusses evaluating trigonometric functions of any angle by using reference angles. It defines trigonometric functions for any real value of the angle θ. For angles greater than 90° or less than 0°, the values can be determined from the acute reference angle. Examples show how to find the reference angle and then use it to evaluate trigonometric functions in different quadrants, including for nonacute angles.
Larson 4.3
This document discusses trigonometric functions and their applications. It defines the six trigonometric functions - sine, cosine, tangent, cotangent, secant, and cosecant - using ratios of sides of a right triangle. Examples are provided to evaluate the trig functions of given angles and to use identities to relate functions. The document also discusses applications of solving right triangles by using trig functions when given angle and side length information.
Larson 4.2
The unit circle relates real numbers to points on a circle of radius 1 centered at the origin. Each real number t corresponds to a point (x, y) on the circle, allowing the definition of the six trigonometric functions in terms of x and y. The trigonometric functions sine and cosine are periodic with period 2π, meaning their values repeat every 2π units. Their domains are all real numbers and their ranges are between -1 and 1.
Larson 4.1
The document describes angles and their measurement in radians and degrees. It defines an angle, measures angles in radians using central angles of a circle, and defines the relationship between radians and degrees. Examples show converting between radians and degrees, finding coterminal angles, complementary and supplementary angles, and calculating arc length and linear speed using angular measurements.
Derivative rules ch2
This document provides the basic derivative rules from Chapter 2. It lists various expressions and their derivatives on subsequent slides, including the derivatives of sums, products, quotients, composite functions, powers, trigonometric functions like sine, cosine, tangent, cotangent, secant and cosecant, and constants. The derivatives covered are: du/dv, df(g(x))/dx, dn/dx, d/dx cot(u), d/dx cu, d/dx tan(u), d/dx (u*v), d/dx c, d/dx cos(u), d/dx sec(u), dx/dx, and d/dx csc(
1.4 Continuity
The document discusses continuity of functions at a point c. It defines continuity as a function being defined at c, the limit existing at c, and the function value at c equaling the limit value. Five examples are given and categorized as continuous, discontinuous with removable discontinuities, or discontinuous with nonremovable discontinuities. Removable discontinuities can be resolved by redefining the function value at c, while nonremovable discontinuities cannot be resolved.
Chapter 5 Definite Integral Topics Review
This document discusses several topics in calculus including RAM theory for approximating functions, computing definite integrals using formulas and a TI-89 calculator, properties of definite integrals like additivity and the mean value theorem, the fundamental theorem of calculus in two parts, and the trapezoidal rule for approximating definite integrals which overestimates for increasing functions and underestimates for decreasing functions.
Lesson 4.3 First and Second Derivative Theory
1) To determine if a function is increasing or decreasing, examine the sign of the first derivative f'(x) at points around any critical points. A change from positive to negative indicates a relative maximum, and a change from negative to positive indicates a relative minimum.
2) A curve is concave up if the second derivative f''(x) is positive, and concave down if f''(x) is negative. To determine concavity, examine the sign of f''(x) around any points where it is zero or undefined. A change in sign indicates an inflection point where the concavity changes.
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Lesson 3.3 3.4 - 1 st and 2nd Derivative Information
Lesson 3.3 3.4 - 1 st and 2nd Derivative Information
Lesson 3.2 Rolle and Mean Value Theorems
- 1. Rolle’s Theorem 1. If f is continuous on [a,b], AND 2. Differentiable on (a,b), AND 3. f(a) = f(b), THEN… …you are guaranteed at least one value c on (a,b) where f’(c) = 0. a b
- 2. Mean Value Theorem 1. If f is continuous on [a,b] AND 2. Differentiable on (a,b), THEN… f ( b) − f ( a ) …there is at least one value c on (a,b) where f ′( c ) = b−a a b