Ch. 5 – Applications of Derivatives 5.2 – Mean Value Theorem.

Slides:

Advertisements

Similar presentations

We Calculus!!! 3.2 Rolle s Theorem and the Mean Value Theorem.

Advertisements

Chapter 3 Application of Derivatives

4.3 Connecting f’ and f’’ with the Graph of f

Section 3.4 – Concavity and the Second Derivative Test

Mean Value Theorem for Derivatives4.2 Teddy Roosevelt National Park, North Dakota Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie.

1 Local Extrema & Mean Value Theorem Local Extrema Rolle’s theorem: What goes up must come down Mean value theorem: Average velocity must be attained Some.

If f (x) is a differentiable function over [ a, b ], then at some point between a and b : Mean Value Theorem for Derivatives.

Maximum and Minimum Value Problems By: Rakesh Biswas

4.2 The Mean Value Theorem.

How Derivatives Affect the Shape of the Graph

12.1 First Derivative and Graph

Chapter 4: Applications of Derivatives Section 4.2: Mean Value Theorem

Mean Value Theorem for Derivatives.

The mileage of a certain car can be approximated by: At what speed should you drive the car to obtain the best gas mileage? Of course, this problem isn’t.

When you see… A1. Find the zeros You think…. A1 To find the zeros...

 Exploration:  Sketch a rectangular coordinate plane on a piece of paper.  Label the points (1, 3) and (5, 3).  Draw the graph of a differentiable.

The Shape of the Graph 3.3. Definition: Increasing Functions, Decreasing Functions Let f be a function defined on an interval I. Then, 1.f increases on.

Review Derivatives When you see the words… This is what you know…  f has a local (relative) minimum at x = a  f(a) is less than or equal to every other.

4.3 How Derivatives Affect the Shape of a Graph. Facts If f ’( x ) > 0 on an interval ( a,b ), then f (x) is increasing on ( a,b ). If f ’( x ) < 0 on.

Applications of Derivatives

A car accelerates from a stop to 45 m/sec in 4 sec. Explain why the car must have been accelerating at exactly m/sec at some moment. 2 Do Now.

In the past, one of the important uses of derivatives was as an aid in curve sketching. We usually use a calculator of computer to draw complicated graphs,

Antiderivatives and uses of derivatives and antiderivatives Ann Newsome.

Section 4.2 Mean Value Theorem What you’ll learn Mean Value Theorem Physical Interpretation Increasing and Decreasing Functions Other Consequences Why?

Ch. 5 – Applications of Derivatives 5.1 – Extreme Values of Functions.

CONCAVITY AND SECOND DERIVATIVE RIZZI – CALC BC. WARM UP Given derivative graph below, find a. intervals where the original function is increasing b.

Ch. 5 – Applications of Derivatives

4.1 Extreme Values of Functions

4.2 Mean Value Theorem Objective SWBAT apply the Mean Value Theorem and find the intervals on which a function is increasing or decreasing.

Increasing & Decreasing Functions & The First Derivative Test (3.3) November 29th, 2012.

Determine where a function is increasing or decreasing When determining if a graph is increasing or decreasing we always start from left and use only the.

4.2 Critical Points Mon Oct 19 Do Now Find the derivative of each 1) 2)

AP Calculus Unit 4 Day 5 Finish Concavity Mean Value Theorem Curve Sketching.

If f (x) is continuous over [ a, b ] and differentiable in (a,b), then at some point, c, between a and b : Mean Value Theorem for Derivatives.

Mean Value Theorem for Derivatives4.2 Teddy Roosevelt National Park, North Dakota Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie.

AP CALCULUS AB FINAL REVIEW APPLICATIONS OF THE DERIVATIVE.

Ch. 5 – Applications of Derivatives 5.1 – Extreme Values of Functions.

Ch. 2 – Limits and Continuity 2.4 – Rates of Change and Tangent Lines.

If f(x) is a continuous function on a closed interval x ∈ [a,b], then f(x) will have both an Absolute Maximum value and an Absolute Minimum value in the.

3.2 Rolle’s Theorem and the

Copyright © Cengage Learning. All rights reserved.

Table of Contents 25. Section 4.3 Mean Value Theorem.

Ch. 2 – Limits and Continuity

Increasing & Decreasing Functions & The First Derivative Test (3.3)

Ch. 5 – Applications of Derivatives

Relating the Graphs of f, f’ and f’’

Table of Contents 21. Section 4.3 Mean Value Theorem.

Learning Target: I will determine if a function is increasing or decreasing and find extrema using the first derivative. Section 3: Increasing & Decreasing.

3.3: Increasing/Decreasing Functions and the First Derivative Test

3.3 Increasing and Decreasing Functions and the First Derivative Test

3.1 Extrema on an Interval Define extrema of a function on an interval. Define relative extrema of a function on an open interval. Find extrema on a closed.

Local Extrema & Mean Value Theorem

First Derivative Test So far…

Increasing and Decreasing Functions and the First Derivative Test

The mileage of a certain car can be approximated by:

3.2 Rolle’s Theorem and the

Applications of Derivatives

4.1 – Extreme Values of Functions

A function f is increasing on an open interval I if, for any choice of x1 and x2 in I, with x1 < x2, we have f(x1) < f(x2). A function f is decreasing.

Section 4.2 Mean value theorem.

5.2 Mean Value Theorem for Derivatives

Warmup 1. What is the interval [a, b] where Rolle’s Theorem is applicable? 2. What is/are the c-values? [-3, 3]

Section 3.4 – Concavity and the Second Derivative Test

4.2 Critical Points, Local Maxima and Local Minima

Rolle’s Theorem and the Mean Value Theorem

Analyzing f(x) and f’(x) /

f(x) g(x) x x (-8,5) (8,4) (8,3) (3,0) (-4,-1) (-7,-1) (3,-2) (0,-3)

Unit 4: Applications of Derivatives

Copyright © Cengage Learning. All rights reserved.

Presentation transcript:

Ch. 5 – Applications of Derivatives 5.2 – Mean Value Theorem

Mean Value Theorem for Derivatives: If f is continuous on a closed interval [a, b], and f is differentiable at every point on the interval, then there exists at least one point such that – This theorem is saying that some x-value from a to b will have a tangent line with a slope equal to the slope of the average slope on the interval! Ex: Find the x-coordinate of the point that satisfies the mean value theorem for f(x) = 2x 2 – 5 on the interval [2, 6]. – Use the formula above! This is the slope of the secant line through the endpoints!!!

A function f is increasing (or decreasing) if its slope is positive (or negative). – If f’(c) > 0, then f is increasing at c – If f’(c) < 0, then f is decreasing at c Ex: Find the local extrema of f(x) = x 2 – x – 12. Also find where the function is increasing and decreasing. – Our only critical point is at x = ½, and there are no endpoints – Make a sign chart to determine where the function is increasing/decreasing... – f is decreasing on (-∞, ½ ) and increasing on (½, ∞) over the domain of f. There is a local minimum at (½, -49/4). + -

Ex: Find the local extrema of. Also find where the function is increasing and decreasing. – Find the critical points... – f‘ = 0 where the numerator is zero, so x = 0 and x = 2 are critical points – Also, f’ doesn’t exist when x = 1, so that’s a critical point – Make a sign chart with all 3 critical points to determine where the function is increasing/decreasing... – f is increasing on (-∞, 0 ) and (2, ∞) and decreasing on (0, 1) and (1, 2) over the domain of f. There is a local minimum at (2, 4) and a local maximum at (0, 0)

Antiderivative: Opposite of a derivative – A function F is an antiderivative of f if F’(x) = f(x) for all x in the domain of f. Ex: Find the antiderivative of the following functions: – Think: which functions gives the following derivatives? – Antiderivative: F(x) = x 2 + C – Antiderivative: F(x) = 4x 4 + C – In general, a function f(x) = x n will have an antiderivative of – Antiderivative: F(x) = -cosx + C The C represents some constant value and must always be included in your answer!

Ex: Find the antiderivative of the following functions: – Antiderivative: F(x) = ln(x + 3) + C – Antiderivative: F(x) = x 6 – 4x 5 + x 2 /2 – 9x + C – Antiderivative: F(x) = 2e x + C Ex: Find f if f’(x) = 3x and f passes through (3, 4). – Find the antiderivative, then solve for C!

Ex: Find the position function of a rock dropped from rest from a 540m cliff with an acceleration of -9.8 m/s 2. – We’ll need to do 2 antiderivatives, one at a time! – Since the object is dropped from rest, v(0) = 0 – Now find s(t)... – The rock is dropped from 540m, so s(t) has the point (0, 540)