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

Mean value theorem example: square root function | AP Calculus AB | Khan Academy
Mean value theorem example: square root function | AP Calculus AB | Khan Academy

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Chapter 4: Applications of Derivatives Section 4.2: Mean Value Theorem
AP CALCULUS AB Chapter 4: Applications of Derivatives Section 4.2: Mean Value Theorem

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What you’ll learn about
Mean Value Theorem Physical Interpretation Increasing and Decreasing Functions Other Consequences …and why The Mean Value Theorem is an important theoretical tool to connect the average and instantaneous rates of change.

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Mean Value Theorem for Derivatives

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This means……. IF y = f(x) is continuous
y = f(x) is on a closed interval [a,b] y = f(x) is differentiable at every point in its interior (a,b) THEN Somewhere between points A and B on a differentiable curve, there is at least one tangent line parallel to chord AB.

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Example 1: Exploring the Mean Value Theorem
Show that the function f(x) = x2 satisfies the hypothesis of the Mean Value Theorem on the interval [0,2]. Then find a solution c to the equation Consider f(x) = x2. Is it continuous? Closed interval? Differentiable? If so, by the MVT we are guaranteed a point c in the interval [0,2] for which

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Example 1 continued Find c Given f(x) = x2 Interval [0,2]
To use the MVT Find the slope of the chord with endpoints (0, f(0)) and (2, f(2)). Find f’ Set f’ equal to the slope of the chord, solve for c Find c

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Interpret your answer The slope of the tangent line to f(x) = x2 at
x = 1 is equal to the slope of the chord AB. OR The tangent line at x = 1 is parallel to chord AB. Write equations for line AB and the tangent line of y = x2 at x=1. Graph and investigate. The lines should be parallel.

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Example Explore the Mean Value Theorem

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Example 2: Mean Value Theorem

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Rolle’s Theorem Rolle’s Theorem:
If f is continuous on the closed interval and differentiable on the open interval and if then there is at least one number c in such that (a, f(a)) (b, f(b) a b

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Show that the function satisfies the hypothesis of the MVT on the interval [0,1]. Then find c
Is it continuous? Closed interval? Differentiable? Find c Interpret your findings

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Example 2: Further Exploration of the MVT
Explain why each of the following functions fails to satisfy the conditions of the Mean Value Theorem on the interval [-1,1]. You try:

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Example 3: Applying the Mean Value Theorem
Find a tangent to f in the interval (-1,1) that is parallel to the secant AB. Given: , A = (-1,f(-1)) and B = (1, f(1)) Find the slope of AB and f’(c) Apply MVT to find c Evaluate f(x) at c, use that point and the slope from step 1 to find the equation of the tangent line

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You Try – Find slope of chord AB that connects endpoints
Find f ’ and apply MVT formula to find c. Evaluate f(x) at c, use that point and the slope from step 1 to find the equation of the tangent line. Graph & check.

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Physical Interpretation of the Mean Value Theorem
The MVT says the instantaneous change at some interior point must equal the average change over the entire interval. f’(x) = instantaneous change at a point = average change over the interval

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Example 4: Interpreting the MVT
If a car accelerating from zero takes 8 sec to go 352 ft, its average velocity for the 8-second interval is 352 / 8 = 44 ft/sec, or 30 mph. Can we cite the driver for speeding if he / she is in a residential area with a speed limit of 25 mph?

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Increasing Function, Decreasing Function

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Corollary: Increasing and Decreasing Functions

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Example Determining Where Graphs Rise or Fall

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Corollary: Functions with f’=0 are Constant

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Section 4.2 – Mean Value Theorem
To find the intervals on which a function is increasing or decreasing Locate the critical numbers of f in Use these numbers to determine the test intervals. (when or is undefined, and the endpoints of the interval). Determine the sign of at a value in each of the test intervals. Decide whether f is increasing or decreasing on each interval. inc dec. f(x) f’(x) a b c

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Section 4.2 – Mean Value Theorem
Ex: inc dec. inc. f(x) + + f’(x) 1/3 1

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Example 5 Determining where graphs rise or fall
Use corollary 1 to determine where the graph of f(x) = x2 – 3x is increasing and decreasing. Find f’ and set it equal to zero to find critical points. Where f ’ > 0, f is increasing. Where f ’ < 0, f is decreasing. Any maximum or minimum values?

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Example 6 Determining where graphs rise or fall
Where is the function increasing and where is it decreasing? Graphically: Use window [-5,5] by [-5,5] Confirm Analytically: Find f ’, evaluate f ’ = 0 to find critical points. Where f ’ > 0, f is increasing Where f ’ < 0, f is decreasing.

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Corollary: Functions with the Same Derivative Differ by a Constant
Example:

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Antiderivative

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Example 7 Applying Corollary 3
Find the function f(x) whose derivative is sin x and whose graph passes through the point (0,2). Write f(x) = antiderivative function + C Use (x,y) = (0, 2) in equation, solve for C. Write f(x), the antiderivative of sin x through (0, 2)

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Find the Antiderivative!
Find the antiderivative of f(x) = 6×2. Reverse the power rule Add 1 to the exponent Divide the coefficient by (exponent + 1) Add C You have found the antiderivative, F(x) Find the antiderivative of f(x) = cos x Think Backwards: What function has a derivative of cos x? Add C You have found the antiderivative, F(x).

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Find each antiderivative – don’t forget C!
f’(x) = -sin x f’(x) = 2x + 6 f’(x) = 2×2 + 4x – 3

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Specific Antiderivatives
Find the antiderivative of through the point (0, 1). Write equation for antiderivative + c Insert point (0,1) for (x, y) and solve for c Write specific antiderivative.

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Example Finding Velocity and Position

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The acceleration is 9.8 m/sec2 and the body is propelled downward with an initial velocity of 1 m/sec2. Work backwards Velocity function + c, P(0,1) (why?) Position function + c (from velocity function)

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Summary The Mean Value Theorem tells us that
If y = f(x) is continuous at every point on the closed interval [a, b] and differentiable at every point of its interior (a,b), then there is at least one point c in (a,b) at which (The derivative at point c = the slope of the chord.) It’s corollaries go on to tell us that where f ‘ is positive, f is increasing, where f ‘ = 0, f is a constant function, and where f ‘ is negative, f is decreasing. We can work backwards from a derivative function to the original function, a process called antidifferentiation. However, as the derivative of any constant = 0, we need to know a point of the original function to get its specific antiderivative. Without a point of the function all we can determine is a general formula for f(x) + C.

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