4.2 - The Mean Value Theorem. Theorems If the conditions (hypotheses) of a theorem are satisfied, the conclusion is known to be true. If the hypotheses of a theorem are not satisfied, the conclusion may still be true, but not guaranteed. Rolle’s Theorem.

By4.2 The Mean Value Theorem. Rolle’s Theorem. Let f be a function that satisfies the following three conditions: f is continuous on the closed interval [a,b] . f is differentiable on the open interval (a,b) . f(a) = f(b) .

ByThe Mean Value Theorem. I wonder how mean this theorem really is?. Lesson 4.2. This is Really Mean. Think About It. Consider a trip of two hours that is 120 miles in distance … You have averaged 60 miles per hour

ByCh 2.4: Differences Between Linear and Nonlinear Equations. Recall that a first order ODE has the form y ' = f ( t , y ), and is linear if f is linear in y, and nonlinear if f is nonlinear in y. Examples: y ' = t y - e t , y ' = t y 2 .

ByInfinite Limits. Lesson 2.5. Previous Mention of Discontinuity. A function can be discontinuous at a point The function goes to infinity at one or both sides of the point, known as a pole Example Enter this function into the Y= screen of your calculator Use standard zoom.

ByLossless Compression in Multimedia Data Representation. Hao Jiang Computer Science Department Sept. 20, 2007. Arithmetic Coding. Arithmetic coding represents a input symbol string as a small interval in [0, 1) The size of the interval equals

ByChapter 9 Maintenance and Replacement . The problem of determining the lifetime of an asset or an activity simultaneously with its management during that lifetime is an important problem in practice. The most typical example is the problem of optimal

BySection 11.1. Vector Valued Functions. Definition of Vector-Valued Function. How are vector valued functions traced out?. In practice it is often easier to rewrite the function. Sketch the curve represented by the vector-valued function and give the orientation of the curve. #26 r (t)=

ByExtrema on an Interval. Lesson 4.1. Design Consultant Problem. A milk company wants to cut down on expenses They decide that their milk carton design uses too much paper For a given volume how can we minimize the amount of paper used?.

ByImportant Theorems about continuous functions. Extreme Value Theorem Intermediate Value Theorem Some applications. Intuitive Picture. Imagine graph of f a 2-dimensional profile of a mountain range Tops of mountains correspond to relative maxima and bottoms of valleys to relative minima

ByI wonder how mean this theorem really is?. The Mean Value Theorem. Lesson 4.2. This is Really Mean. Think About It. Consider a trip of two hours that is 120 miles in distance … You have averaged 60 miles per hour

BySolving Inequalities. We can solve inequalities just like equations, with the following exception: Multiplication or division of an inequality by a negative number reverses the direction of the inequality. Problem. Solve 2 x + 11 5 x – 1.

ByLearning Objectives for Section 10.4 The Derivative . The student will be able to calculate rate of change and slope of the tangent line. The student will be able to interpret the meaning of the derivative. The student will be able to identify the nonexistence of the derivative.

ByAssignment 4. Section 3.1 The Derivative and Tangent Line Problem. The Basic Question is…. How do you find the equation of a line that is tangent to a function y=f(x) at an arbitrary point P? To find the equation of a line you need: a point and a slope.

By10.3: Continuity. Definition of Continuity . A function f is continuous at a point x = c if 1. 2. f ( c ) exists 3. A function f is continuous on the open interval ( a , b ) if it is continuous at each point on the interval.

ByConcavity and the 2nd Derivative Test. Objectives: Be able to determine where a function is concave upward or concave downward with the use of calculus. Be able to apply the second derivative test to find the relative extrema of a function. Critical Vocabulary:

By4.1 Extreme Values of Functions. Greg Kelly, Hanford High School Richland, Washington. The mileage of a certain car can be approximated by:. At what speed should you drive the car to obtain the best gas mileage?. Look at the following example :.

By4.3 Extreme Values of Functions. Borax Mine, Boron, CA Photo by Vickie Kelly, 2004. Greg Kelly, Hanford High School, Richland, Washington. Local Extreme Values:. A local maximum is the maximum value within some open interval.

By4.1 Extreme Values of Functions. Greg Kelly, Hanford High School Richland, Washington. The mileage of a certain car can be approximated by:. At what speed should you drive the car to obtain the best gas mileage?. The textbook gives the following example at the start of chapter 4:.

ByRolle's Theorem. Objectives: Be able to determine if Rolle’s Theorem can be applied to a function on a closed interval. 2. Be able to apply Rolle’s Theorem to various functions. Critical Vocabulary: Rolle’s Theorem.

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