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Calculus Concepts Overview

Understanding extrema, Mean Value Theorem, increasing/decreasing functions, antiderivatives, and related examples in calculus. Explore functions, derivatives, and interpretations.

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Calculus Concepts Overview

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  1. DO NOW: Find C so that the graph of the function f passes through the specified point. f(x) = -2x + C, (-2,7) g(x) = x2 + 2x + C, (1,-1)

  2. QUICK REVIEW: • How do you find extrema? • Find all extrema of

  3. 2 Theorems we learned about so far…

  4. 4.2 - Mean Value Theorem HW: Pg. 3-33 multiples of 3, 35, 37, 40, 43, 54

  5. Mean Value Theorem • If y = f(x) is continuous at every point of 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

  6. Does f(x) = x2 satisfy the hypotheses of the Mean Value Theorem on the interval [0,3]

  7. Example 2Why do the following functions fail to satisfy the Mean Value Theorem on the interval [-1,1]? For x < 1 For x ≥ 1

  8. Example 2 (solution)

  9. Example 3 • Let f(x) = √(1 - x2), A = (-1, f(-1)), and B = (1, f(1)). Find a tangent of f in the interval (-1,1) that is parallel to the secant AB.

  10. Interpretation of the Mean Value Theorem • If a Ferrari accelerating from zero takes 4 seconds to go 356 ft, its average velocity for the 4 second interval is 356/4 = 89 ft/sec or 60 mph • The theorem states that at some point while the Ferrari is accelerating, the speedometer must read exactly 60 mph.

  11. DEFINITIONS • INCREASING FUNCTION, DECREASING FUNCTION • Let f be a function defined on an interval I and let x1 and x2 be any two points in I. • f increases on I if x1 < x2 => f(x1) < f(x2) • f decreases on I if x1 < x2 => f(x1) > f(x2)

  12. Increasing and Decreasing Functions • Let f be continuous on [a,b] and differentiable on (a,b) • If f’ > 0 at each point of (a,b), then f increases on [a,b] • If f’ < 0 at each point of (a,b), then f decreases on [a,b]

  13. Example 5 • Where is the function f(x) = x3 – 5x increasing and where is it decreasing?

  14. Functions with f’ = 0 are constants • If f’(x) = 0 at each point of an interval I, then there is a constant C where f(x) = C for all x in I. • Example:

  15. Also… Functions with the Same Derivative Differ by a Constant If f’(x) = g’(x) at each point of an interval I, then there is a constant C such that f(x) = g(x) + C for all x in I. Example:

  16. Example 6 • Find the function f(x) whose derivative is sinx and whose graph passes through the point (0,2).

  17. Antiderivative • A function F(x) is an antiderivatve of a function f(x) if F’(x) = f(x) for all x in the domain of f. The process of finding an antiderivative is antidifferentiation.

  18. Example 7Find the velocity and Position • Find the velocity and position functions of a body falling freely from a height of 0 meters under each of the following sets of conditions: • The acceleration is 9.8 m/sec2 and the body falls from rest. • The acceleration is 9.8 m/sec2 and the body is propelled downward with an initial velocity of 1 m/sec.

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