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4.5 continuous functions and differentiable functions

The document discusses continuous and differentiable functions. It defines elementary functions as those constructed using basic operations like addition and multiplication. Continuous functions over a closed interval are bounded and have absolute maximum and minimum values. The Intermediate Value Theorem states that a continuous function takes on all values between its minimum and maximum. Differentiable functions are continuous. Rolle's Theorem says that if a differentiable function is equal at the endpoints of an interval, its derivative is zero somewhere in between.

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Continuous and Differentiable Functions
Continuous and Differentiable Functions In this section we highlight some important facts about continuous functions and differentiable functions.
Continuous and Differentiable Functions In this section we highlight some important facts about continuous functions and differentiable functions. Elementary Functions
Continuous and Differentiable Functions In this section we highlight some important facts about continuous functions and differentiable functions. Elementary Functions Recall that a formula that may be constructed from the “basic formulas” by applying the +, – , *, / and the composition operations in finitely many steps is called an elementary formula.
Continuous and Differentiable Functions In this section we highlight some important facts about continuous functions and differentiable functions. Elementary Functions Recall that a formula that may be constructed from the “basic formulas” by applying the +, – , *, / and the composition operations in finitely many steps is called an elementary formula. Most commonly used formulas are elementary formulas.
Continuous and Differentiable Functions In this section we highlight some important facts about continuous functions and differentiable functions. Elementary Functions Recall that a formula that may be constructed from the “basic formulas” by applying the +, – , *, / and the composition operations in finitely many steps is called an elementary formula. Most commonly used formulas are elementary formulas. These formulas are also the ones whose derivatives can be computed easily
Continuous and Differentiable Functions In this section we highlight some important facts about continuous functions and differentiable functions. Elementary Functions Recall that a formula that may be constructed from the “basic formulas” by applying the +, – , *, / and the composition operations in finitely many steps is called an elementary formula. Most commonly used formulas are elementary formulas. These formulas are also the ones whose derivatives can be computed easily. But under this definition, one may constructed an elementary function whose discontinuity is quite messy. http://mathoverflow.net/questions/17901/exis tence-of-antiderivatives-of-nasty-but- For a heated discussion: elementary-functions
Continuous and Differentiable Functions In this section we highlight some important facts about continuous functions and differentiable functions. Elementary Functions Recall that a formula that may be constructed from the “basic formulas” by applying the +, – , *, / and the composition operations in finitely many steps is called an elementary formula. Most commonly used formulas are elementary formulas. These formulas are also the ones whose derivatives can be computed easily. But under this definition, one may constructed an elementary function whose discontinuity is quite messy. However there are theorems about the continuous functions and the differentiable functions which we may apply to elementary functions.
Continuous and Differentiable Functions Continuous Functions
Continuous and Differentiable Functions Continuous Functions
Continuous and Differentiable Functions Continuous Functions There are two important facts about continuous functions.
Continuous and Differentiable Functions Continuous Functions There are two important facts about continuous functions. I. Intermediate Value Theorem
Continuous and Differentiable Functions Continuous Functions There are two important facts about continuous functions. I. Intermediate Value Theorem Let f(x) be a continuous function defined over the closed interval [a, b] such that f(a) < f(b),
Continuous and Differentiable Functions Continuous Functions There are two important facts about continuous functions. I. Intermediate Value Theorem Let f(x) be a continuous function defined over the closed interval [a, b] such that f(a) < f(b), f(b) f(a) a b
Continuous and Differentiable Functions Continuous Functions There are two important facts about continuous functions. I. Intermediate Value Theorem Let f(x) be a continuous function defined over the closed interval [a, b] such that f(a) < f(b), let m be any number where f(a) < m < f(b), f(b) m f(a) a b
Continuous and Differentiable Functions Continuous Functions There are two important facts about continuous functions. I. Intermediate Value Theorem Let f(x) be a continuous function defined over the closed interval [a, b] such that f(a) < f(b), let m be any number where f(a) < m < f(b), then there exists at least one c, i.e. one or more, f(b) where a < c < b, m f(a) a c b
Continuous and Differentiable Functions Continuous Functions There are two important facts about continuous functions. I. Intermediate Value Theorem Let f(x) be a continuous function defined over the closed interval [a, b] such that f(a) < f(b), let m be any number where f(a) < m < f(b), then there exists at least one c, i.e. one or more, f(b) where a < c < b, m and that f(c) = m. f(a) a c b
Continuous and Differentiable Functions Continuous Functions There are two important facts about continuous functions. I. Intermediate Value Theorem Let f(x) be a continuous function defined over the closed interval [a, b] such that f(a) < f(b), let m be any number where f(a) < m < f(b), then there exists at least other c’s one c, i.e. one or more, f(b) where a < c < b, m and that f(c) = m. f(a) a c b
Continuous and Differentiable Functions Continuous Functions There are two important facts about continuous functions. I. Intermediate Value Theorem Let f(x) be a continuous function defined over the closed interval [a, b] such that f(a) < f(b), let m be any number where f(a) < m < f(b), then there exists at least other c’s one c, i.e. one or more, f(b) where a < c < b, m and that f(c) = m. We omit the proof here. f(a) Here is a link to its proof. http://en.wikipedia.org/wiki/Intermediate a c b _value_theorem
Continuous and Differentiable Functions Remarks I. Similarly, if f(a) > f(b) then there is some c with a < c < b such that f(c) = m for f(a) > m > f(b).
Continuous and Differentiable Functions Remarks I. Similarly, if f(a) > f(b) then there is some c with a < c < b such that f(c) = m for f(a) > m > f(b).
Continuous and Differentiable Functions Remarks I. Similarly, if f(a) > f(b) then there is some c with a < c < b such that f(c) = m for f(a) > m > f(b). II. If the condition is f(a) ≤ m ≤ f(b) then the conclusion is a ≤ c ≤ b.
Continuous and Differentiable Functions Remarks I. Similarly, if f(a) > f(b) then there is some c with a < c < b such that f(c) = m for f(a) > m > f(b). II. If the condition is f(a) ≤ m ≤ f(b) then the conclusion is a ≤ c ≤ b. One important application for this theorem is the existence of roots. (Bozano’s Theorem) Let y = f(x) be a continuous function over the closed interval [a, b], then there exists at least one c where a < c < b with f(c) = 0 if the signs of f(a) and f(b) are different.
Continuous and Differentiable Functions Remarks I. Similarly, if f(a) > f(b) then there is some c with a < c < b such that f(c) = m for f(a) > m > f(b). II. If the condition is f(a) ≤ m ≤ f(b) then the conclusion is a ≤ c ≤ b. One important application for this theorem is the existence of roots. (Bozano’s Theorem) Let y = f(x) be a continuous function over the closed interval [a, b], then there exists at least one c where a < c < b with f(c) = 0 if the signs of f(a) and f(b) are different. With this theorem, we conclude that f(x) = x3 – 3x2 – 5 has a root between 2 < x < 5, as in 3.6 Example B, because f(2) and f(5) have opposite signs.
Continuous and Differentiable Functions The other important fact about continuous functions is the existence of extrema over a closed interval.
Continuous and Differentiable Functions The other important fact about continuous functions is the existence of extrema over a closed interval. II. Extrema Theorem for Continuous Functions
Continuous and Differentiable Functions The other important fact about continuous functions is the existence of extrema over a closed interval. II. Extrema Theorem for Continuous Functions Let y = f(x) be a continuous function defined over a closed interval V = [a, b], then both the absolute max. and the absolute min. exist in V.
Continuous and Differentiable Functions The other important fact about continuous functions is the existence of extrema over a closed interval. II. Extrema Theorem for Continuous Functions Let y = f(x) be a continuous function defined over a closed interval V = [a, b], then both the absolute max. and the absolute min. exist in V. In particular if M is the maximum and m is the minimum, then m ≤ f(x) ≤ M for every x in the interval [a, b].
Continuous and Differentiable Functions The other important fact about continuous functions is the existence of extrema over a closed interval. II. Extrema Theorem for Continuous Functions Let y = f(x) be a continuous function defined over a closed interval V = [a, b], then both the absolute max. and the absolute min. exist in V. In particular if M is the maximum and m is the minimum, then m ≤ f(x) ≤ M for every x in the interval [a, b]. So a continuous function defined over a closed interval V is always bounded.
Continuous and Differentiable Functions The other important fact about continuous functions is the existence of extrema over a closed interval. II. Extrema Theorem for Continuous Functions Let y = f(x) be a continuous function defined over a closed interval V = [a, b], then both the absolute max. and the absolute min. exist in V. In particular if M is the maximum and m is the minimum, then m ≤ f(x) ≤ M for every x in the interval [a, b]. So a continuous function defined over a closed interval V is always bounded. Corollary. Let y = f(x) be an elementary function defined over a closed interval V = [a, b], then f(x) is continuous over [a, b], hence both the absolute max. and the absolute min. exist in V.
Continuous and Differentiable Functions Differentiable Functions
Continuous and Differentiable Functions Differentiable Functions We state the following important theorems about differentiable functions without proofs.
Continuous and Differentiable Functions Differentiable Functions We state the following important theorems about differentiable functions without proofs. Differentiability is a lot stronger condition than continuity at a point on a graph.
Continuous and Differentiable Functions Differentiable Functions We state the following important theorems about differentiable functions without proofs. Differentiability is a lot stronger condition than continuity at a point on a graph. Theorem. If f(x) is differentiable at x = a then it is continuous at x = a.
Continuous and Differentiable Functions Differentiable Functions We state the following important theorems about differentiable functions without proofs. Differentiability is a lot stronger condition than continuity at a point on a graph. Theorem. If f(x) is differentiable at x = a then it is continuous at x = a. Differentiability means the rate of change may be measured. This observation leads to Rolle’s Theorem which gives the existence of a point c where f'(c) = 0.
Continuous and Differentiable Functions Differentiable Functions We state the following important theorems about differentiable functions without proofs. Differentiability is a lot stronger condition than continuity at a point on a graph. Theorem. If f(x) is differentiable at x = a then it is continuous at x = a. Differentiability means the rate of change may be measured. The rate of change at an extremum x = c of f(x) must be 0 because it can’t be + (increasing) or – (decreasing) hence f'(c) = 0.
Continuous and Differentiable Functions Differentiable Functions We state the following important theorems about differentiable functions without proofs. Differentiability is a lot stronger condition than continuity at a point on a graph. Theorem. If f(x) is differentiable at x = a then it is continuous at x = a. Differentiability means the rate of change may be measured. The rate of change at an extremum x = c of f(x) must be 0 because it can’t be + (increasing) or – (decreasing) hence f'(c) = 0. This observation leads to Rolle’s Theorem which gives the existence of a point c where f'(c) = 0.
Continuous and Differentiable Functions Rolle’s Theorem Let f(x) be a differentiable function defined over the closed interval [a, b] with a < b and that f(a) = f(b),
Continuous and Differentiable Functions Rolle’s Theorem Let f(x) be a differentiable function defined over the closed interval [a, b] with a < b and that f(a) = f(b), f(a)=f(b) x a c b x
Continuous and Differentiable Functions Rolle’s Theorem Let f(x) be a differentiable function defined over the closed interval [a, b] with a < b and that f(a) = f(b), then there is at least one c where a < c < b such that f'(c) = 0. other c’s f'(c) = 0 f(a)=f(b) x a c b x
Continuous and Differentiable Functions Rolle’s Theorem Let f(x) be a differentiable function defined over the closed interval [a, b] with a < b and that f(a) = f(b), then there is at least one c where a < c < b such that f'(c) = 0. other c’s Proof. Consider the following f'(c) = 0 two cases. f(a)=f(b) x a c b x
Continuous and Differentiable Functions Rolle’s Theorem Let f(x) be a differentiable function defined over the closed interval [a, b] with a < b and that f(a) = f(b), then there is at least one c where a < c < b such that f'(c) = 0. other c’s Proof. Consider the following f'(c) = 0 two cases. 1. The function f(x) is a f(a)=f(b) constant function, i.e. x a c b x f(x) = f(a) = k, then f'(x) = 0
Continuous and Differentiable Functions Rolle’s Theorem Let f(x) be a differentiable function defined over the closed interval [a, b] with a < b and that f(a) = f(b), then there is at least one c where a < c < b such that f'(c) = 0. other c’s Proof. Consider the following f'(c) = 0 two cases. 1. The function f(x) is a f(a)=f(b) constant function, i.e. x a c b x f(x) = f(a) = k, then f'(x) = 0. Any number c where a < c < b would satisfy f'(c) = 0.
Continuous and Differentiable Functions Rolle’s Theorem Let f(x) be a differentiable function defined over the closed interval [a, b] with a < b and that f(a) = f(b), then there is at least one c where a < c < b such that f'(c) = 0. other c’s Proof. Consider the following f'(c) = 0 two cases. 1. The function f(x) is a f(a)=f(b) constant function, i.e. x a c b x f(x) = f(a) = k, then f'(x) = 0 Any number c where a < c < b would satisfy f'(c) = 0. 2. If the function f(x) is not a constant function, then there exists an extremum c between a and b, with f(c) ≠ f(a) and f(c) ≠ (b) and we must have f'(c) = 0.
Continuous and Differentiable Functions The average rate of change of y = f(x) from x = a to b is f(b) – f(a) = Δy = slope of the chord as shown. Δx b–a Avg. rate of change y = f(x) = slope of the chord (b, f(b)) Δy = Δx Δy Δx (a, f(a)) a b
Continuous and Differentiable Functions The average rate of change of y = f(x) from x = a to b is f(b) – f(a) = Δy = slope of the chord as shown. Δx b–a The graph y = f(x) is the rotation of the graph of some function y = g(x) defined over some interval [A, B] Avg. rate of change y = g(x) y = f(x) = slope of the chord (b, f(b)) g(A) = g(B) Δy = Δx (A, g(A)) (B, g(B)) Δy Δx (a, f(a)) Rotate A C B a b
Continuous and Differentiable Functions The average rate of change of y = f(x) from x = a to b is f(b) – f(a) = Δy = slope of the chord as shown. Δx b–a The graph y = f(x) is the rotation of the graph of some function y = g(x) defined over some interval [A, B] with g(A) = g(B) with the rotation taking (A,g(A)) to (a, f(a)) and (B,g(B)) to (b, f(b)) as shown. Avg. rate of change y = g(x) y = f(x) = slope of the chord (b, f(b)) g(A) = g(B) Δy = Δx (A, g(A)) (B, g(B)) Δy Δx (a, f(a)) Rotate A C B a b
Continuous and Differentiable Functions If in addition y = g(x) is differentiable over an interval that contains [A, B], then Rolle’s Theorem implies the existence of at least one C where A < C < B such that the tangent line at (C, g(C)) is a horizontal. y = f(x) Rolle’s Theorem y = g(x) (b, f(b)) g(A) = g(B) (A, g(A)) (B, g(B)) (a, f(a)) There exists a C where g'(C) = 0 Rotate A C B a b
Continuous and Differentiable Functions If in addition y = g(x) is differentiable over an interval that contains [A, B], then Rolle’s Theorem implies the existence of at least one C where A < C < B such that the tangent line at (C, g(C)) is a horizontal. Under the rotation this horizontal line rotates into a tangent line that is parallel to the chord from (a, f(a)) to (b, f(b)) y = f(x) Rolle’s Theorem y = g(x) (b, f(b)) g(A) = g(B) (A, g(A)) (B, g(B)) (a, f(a)) There exists a C There exists a C where where g'(C) = 0 Rotate f '(C) = Avg. rate of change A C B a b
Continuous and Differentiable Functions If in addition y = g(x) is differentiable over an interval that contains [A, B], then Rolle’s Theorem implies the existence of at least one C where A < C < B such that the tangent line at (C, g(C)) is a horizontal. Under the rotation this horizontal line rotates into a tangent line that is parallel to the chord from (a, f(a)) to (b, f(b)) which gives us the Mean Value Theorem. y = f(x) Rolle’s Theorem Mean Value Theorem y = g(x) (b, f(b)) g(A) = g(B) (A, g(A)) (B, g(B)) (a, f(a)) There exists a C There exists a C where where g'(C) = 0 Rotate f '(C) = Avg. rate of change A C B a b
Continuous and Differentiable Functions Mean Value Theorem Let y = f(x) be a differentiable function over an interval that contains [a, b], then there is at least one c where a < c < b such that y = f(x) f '(c) = f(b) – f(a) . slope = Avg. rate of change (b, f(b)) b–a = f(b) – f(a) b–a . (a, f(a)) a c b There exists a c where f(b) – f(a) f '(c) = Avg. rate of change = b–a .
Continuous and Differentiable Functions Mean Value Theorem Let y = f(x) be a differentiable function over an interval that contains [a, b], then there is at least one c where a < c < b such that y = f(x) f '(c) = f(b) – f(a) . slope = Avg. rate of change (b, f(b)) b–a = f(b) – f(a) b–a . Remarks 1. The precise condition for the theorem is that (a, f(a)) f(x) is continuous in [a, b], and differentiable in (a, b). The condition above is stronger but is sufficient a c b for our purposes. There exists a c where f(b) – f(a) f '(c) = Avg. rate of change = b–a .
Continuous and Differentiable Functions 2. In statistics, the word “Mean” denotes the “average”. However, the statement of, the “Mean Value” in the Mean Value Theorem refers to the average value of the rate–of–change, y = f(x) slope = Avg. rate of change (b, f(b)) f(b) – f(a) = b–a . (a, f(a)) a c b There exists a c where f(b) – f(a) f '(c) = Avg. rate of change = b–a .
Continuous and Differentiable Functions 2. In statistics, the word “Mean” denotes the “average”. However, the statement of, the “Mean Value” in the Mean Value Theorem refers to the average value of the rate–of–change, y = f(x) not the “mean value” slope = Avg. rate of change (b, f(b)) f(b) – f(a) or the average value of = b–a . the function f(x) over the interval [a, b]. (a, f(a)) a c b There exists a c where f(b) – f(a) f '(c) = Avg. rate of change = b–a .
Continuous and Differentiable Functions 2. In statistics, the word “Mean” denotes the “average”. However, the statement of, the “Mean Value” in the Mean Value Theorem refers to the average value of the rate–of–change, y = f(x) not the “mean value” slope = Avg. rate of change (b, f(b)) f(b) – f(a) or the average value of = b–a . the function f(x) over the interval [a, b]. The average value of (a, f(a)) the function f(x) over the interval [a, b] is defined by integrals which are c our next topics. a b There exists a c where f(b) – f(a) f '(c) = Avg. rate of change = b–a .
Continuous and Differentiable Functions Example A. We left Los Angeles at 12:00 PM arrived at San Francisco at 3:30 PM that same afternoon covering a distance of 350 miles.
Continuous and Differentiable Functions Example A. We left Los Angeles at 12:00 PM arrived at San Francisco at 3:30 PM that same afternoon covering a distance of 350 miles. Therefore our average rate is 350 / 3½ miles/hr = 100 mph.
Continuous and Differentiable Functions Example A. We left Los Angeles at 12:00 PM arrived at San Francisco at 3:30 PM that same afternoon covering a distance of 350 miles. Therefore our average rate is 350 / 3½ miles/hr = 100 mph. By the Mean Value Theorem, our speedometer must display the speed at exactly 100mph at some point in time during our trip.
Continuous and Differentiable Functions Example A. We left Los Angeles at 12:00 PM arrived at San Francisco at 3:30 PM that same afternoon covering a distance of 350 miles. Therefore our average rate is 350 / 3½ miles/hr = 100 mph. By the Mean Value Theorem, our speedometer must display the speed at exactly 100mph at some point in time during our trip. Example B. Find the x and y intercepts of the line that is tangent to the graph y = √2x + 1 and is parallel to the chord connecting the points with x = 0 and x = 4. Use a graphing software to draw the graph to confirm your answers.
Continuous and Differentiable Functions Example A. We left Los Angeles at 12:00 PM arrived at San Francisco at 3:30 PM that same afternoon covering a distance of 350 miles. Therefore our average rate is 350 / 3½ miles/hr = 100 mph. By the Mean Value Theorem, our speedometer must display the speed at exactly 100mph at some point in time during our trip. Example B. Find the x and y intercepts of the line that is tangent to the graph y = √2x + 1 and is parallel to the chord connecting the points with x = 0 and x = 4. Use a graphing software to draw the graph to confirm your answers. The end points in question are (0, 1) and (4, 3) so the chord connecting them has slope ½.
Continuous and Differentiable Functions The function y = √2x + 1 is differentiable and satisfies the condition of the Mean Value Theorem, therefore there is some x = c where 0 < c < 4 and y'(c) = ½ .
Continuous and Differentiable Functions The function y = √2x + 1 is differentiable and satisfies the condition of the Mean Value Theorem, therefore there is some x = c where 0 < c < 4 and y'(c) = ½ . We may solve for c directly.
Continuous and Differentiable Functions The function y = √2x + 1 is differentiable and satisfies the condition of the Mean Value Theorem, therefore there is some x = c where 0 < c < 4 and y'(c) = ½ . We may solve for c directly. y = √2x + 1 → y'(x) = 1/√2x + 1
Continuous and Differentiable Functions The function y = √2x + 1 is differentiable and satisfies the condition of the Mean Value Theorem, therefore there is some x = c where 0 < c < 4 and y'(c) = ½ . We may solve for c directly. y = √2x + 1 → y'(x) = 1/√2x + 1 Set 1 = 1 √2x + 1 2
Continuous and Differentiable Functions The function y = √2x + 1 is differentiable and satisfies the condition of the Mean Value Theorem, therefore there is some x = c where 0 < c < 4 and y'(c) = ½ . We may solve for c directly. y = √2x + 1 → y'(x) = 1/√2x + 1 Set 1 = 1 √2x + 1 2 √2x + 1 = 2
Continuous and Differentiable Functions The function y = √2x + 1 is differentiable and satisfies the condition of the Mean Value Theorem, therefore there is some x = c where 0 < c < 4 and y'(c) = ½ . We may solve for c directly. y = √2x + 1 → y'(x) = 1/√2x + 1 Set 1 = 1 √2x + 1 2 √2x + 1 = 2 2x + 1 = 4 x = 3/2
Continuous and Differentiable Functions The function y = √2x + 1 is differentiable and satisfies the condition of the Mean Value Theorem, therefore there is some x = c where 0 < c < 4 and y'(c) = ½ . We may solve for c directly. y = √2x + 1 → y'(x) = 1/√2x + 1 Set 1 = 1 √2x + 1 2 √2x + 1 = 2 2x + 1 = 4 x = 3/2 Therefore the tangent line passes through (3/2, 2) and it has slope ½.
Continuous and Differentiable Functions The function y = √2x + 1 is differentiable and satisfies the condition of the Mean Value Theorem, therefore there is some x = c where 0 < c < 4 and y'(c) = ½ . We may solve for c directly. y = √2x + 1 → y'(x) = 1/√2x + 1 Set 1 = 1 √2x + 1 2 √2x + 1 = 2 2x + 1 = 4 x = 3/2 Therefore the tangent line passes through (3/2, 2) and it has slope ½. So the equation of the tangent line in question is y = ½ (x – 3/2) + 2 or y = x/2 + 5/4. Its x intercept is at –5/2 and the y intercept is at 5/4.

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4.5 continuous functions and differentiable functions

  • 1.
  • 2.
    Continuous and DifferentiableFunctions In this section we highlight some important facts about continuous functions and differentiable functions.
  • 3.
    Continuous and DifferentiableFunctions In this section we highlight some important facts about continuous functions and differentiable functions. Elementary Functions
  • 4.
    Continuous and DifferentiableFunctions In this section we highlight some important facts about continuous functions and differentiable functions. Elementary Functions Recall that a formula that may be constructed from the “basic formulas” by applying the +, – , *, / and the composition operations in finitely many steps is called an elementary formula.
  • 5.
    Continuous and DifferentiableFunctions In this section we highlight some important facts about continuous functions and differentiable functions. Elementary Functions Recall that a formula that may be constructed from the “basic formulas” by applying the +, – , *, / and the composition operations in finitely many steps is called an elementary formula. Most commonly used formulas are elementary formulas.
  • 6.
    Continuous and DifferentiableFunctions In this section we highlight some important facts about continuous functions and differentiable functions. Elementary Functions Recall that a formula that may be constructed from the “basic formulas” by applying the +, – , *, / and the composition operations in finitely many steps is called an elementary formula. Most commonly used formulas are elementary formulas. These formulas are also the ones whose derivatives can be computed easily
  • 7.
    Continuous and DifferentiableFunctions In this section we highlight some important facts about continuous functions and differentiable functions. Elementary Functions Recall that a formula that may be constructed from the “basic formulas” by applying the +, – , *, / and the composition operations in finitely many steps is called an elementary formula. Most commonly used formulas are elementary formulas. These formulas are also the ones whose derivatives can be computed easily. But under this definition, one may constructed an elementary function whose discontinuity is quite messy. http://mathoverflow.net/questions/17901/exis tence-of-antiderivatives-of-nasty-but- For a heated discussion: elementary-functions
  • 8.
    Continuous and DifferentiableFunctions In this section we highlight some important facts about continuous functions and differentiable functions. Elementary Functions Recall that a formula that may be constructed from the “basic formulas” by applying the +, – , *, / and the composition operations in finitely many steps is called an elementary formula. Most commonly used formulas are elementary formulas. These formulas are also the ones whose derivatives can be computed easily. But under this definition, one may constructed an elementary function whose discontinuity is quite messy. However there are theorems about the continuous functions and the differentiable functions which we may apply to elementary functions.
  • 9.
    Continuous and DifferentiableFunctions Continuous Functions
  • 10.
    Continuous and DifferentiableFunctions Continuous Functions
  • 11.
    Continuous and DifferentiableFunctions Continuous Functions There are two important facts about continuous functions.
  • 12.
    Continuous and DifferentiableFunctions Continuous Functions There are two important facts about continuous functions. I. Intermediate Value Theorem
  • 13.
    Continuous and DifferentiableFunctions Continuous Functions There are two important facts about continuous functions. I. Intermediate Value Theorem Let f(x) be a continuous function defined over the closed interval [a, b] such that f(a) < f(b),
  • 14.
    Continuous and DifferentiableFunctions Continuous Functions There are two important facts about continuous functions. I. Intermediate Value Theorem Let f(x) be a continuous function defined over the closed interval [a, b] such that f(a) < f(b), f(b) f(a) a b
  • 15.
    Continuous and DifferentiableFunctions Continuous Functions There are two important facts about continuous functions. I. Intermediate Value Theorem Let f(x) be a continuous function defined over the closed interval [a, b] such that f(a) < f(b), let m be any number where f(a) < m < f(b), f(b) m f(a) a b
  • 16.
    Continuous and DifferentiableFunctions Continuous Functions There are two important facts about continuous functions. I. Intermediate Value Theorem Let f(x) be a continuous function defined over the closed interval [a, b] such that f(a) < f(b), let m be any number where f(a) < m < f(b), then there exists at least one c, i.e. one or more, f(b) where a < c < b, m f(a) a c b
  • 17.
    Continuous and DifferentiableFunctions Continuous Functions There are two important facts about continuous functions. I. Intermediate Value Theorem Let f(x) be a continuous function defined over the closed interval [a, b] such that f(a) < f(b), let m be any number where f(a) < m < f(b), then there exists at least one c, i.e. one or more, f(b) where a < c < b, m and that f(c) = m. f(a) a c b
  • 18.
    Continuous and DifferentiableFunctions Continuous Functions There are two important facts about continuous functions. I. Intermediate Value Theorem Let f(x) be a continuous function defined over the closed interval [a, b] such that f(a) < f(b), let m be any number where f(a) < m < f(b), then there exists at least other c’s one c, i.e. one or more, f(b) where a < c < b, m and that f(c) = m. f(a) a c b
  • 19.
    Continuous and DifferentiableFunctions Continuous Functions There are two important facts about continuous functions. I. Intermediate Value Theorem Let f(x) be a continuous function defined over the closed interval [a, b] such that f(a) < f(b), let m be any number where f(a) < m < f(b), then there exists at least other c’s one c, i.e. one or more, f(b) where a < c < b, m and that f(c) = m. We omit the proof here. f(a) Here is a link to its proof. http://en.wikipedia.org/wiki/Intermediate a c b _value_theorem
  • 20.
    Continuous and DifferentiableFunctions Remarks I. Similarly, if f(a) > f(b) then there is some c with a < c < b such that f(c) = m for f(a) > m > f(b).
  • 21.
    Continuous and DifferentiableFunctions Remarks I. Similarly, if f(a) > f(b) then there is some c with a < c < b such that f(c) = m for f(a) > m > f(b).
  • 22.
    Continuous and DifferentiableFunctions Remarks I. Similarly, if f(a) > f(b) then there is some c with a < c < b such that f(c) = m for f(a) > m > f(b). II. If the condition is f(a) ≤ m ≤ f(b) then the conclusion is a ≤ c ≤ b.
  • 23.
    Continuous and DifferentiableFunctions Remarks I. Similarly, if f(a) > f(b) then there is some c with a < c < b such that f(c) = m for f(a) > m > f(b). II. If the condition is f(a) ≤ m ≤ f(b) then the conclusion is a ≤ c ≤ b. One important application for this theorem is the existence of roots. (Bozano’s Theorem) Let y = f(x) be a continuous function over the closed interval [a, b], then there exists at least one c where a < c < b with f(c) = 0 if the signs of f(a) and f(b) are different.
  • 24.
    Continuous and DifferentiableFunctions Remarks I. Similarly, if f(a) > f(b) then there is some c with a < c < b such that f(c) = m for f(a) > m > f(b). II. If the condition is f(a) ≤ m ≤ f(b) then the conclusion is a ≤ c ≤ b. One important application for this theorem is the existence of roots. (Bozano’s Theorem) Let y = f(x) be a continuous function over the closed interval [a, b], then there exists at least one c where a < c < b with f(c) = 0 if the signs of f(a) and f(b) are different. With this theorem, we conclude that f(x) = x3 – 3x2 – 5 has a root between 2 < x < 5, as in 3.6 Example B, because f(2) and f(5) have opposite signs.
  • 25.
    Continuous and DifferentiableFunctions The other important fact about continuous functions is the existence of extrema over a closed interval.
  • 26.
    Continuous and DifferentiableFunctions The other important fact about continuous functions is the existence of extrema over a closed interval. II. Extrema Theorem for Continuous Functions
  • 27.
    Continuous and DifferentiableFunctions The other important fact about continuous functions is the existence of extrema over a closed interval. II. Extrema Theorem for Continuous Functions Let y = f(x) be a continuous function defined over a closed interval V = [a, b], then both the absolute max. and the absolute min. exist in V.
  • 28.
    Continuous and DifferentiableFunctions The other important fact about continuous functions is the existence of extrema over a closed interval. II. Extrema Theorem for Continuous Functions Let y = f(x) be a continuous function defined over a closed interval V = [a, b], then both the absolute max. and the absolute min. exist in V. In particular if M is the maximum and m is the minimum, then m ≤ f(x) ≤ M for every x in the interval [a, b].
  • 29.
    Continuous and DifferentiableFunctions The other important fact about continuous functions is the existence of extrema over a closed interval. II. Extrema Theorem for Continuous Functions Let y = f(x) be a continuous function defined over a closed interval V = [a, b], then both the absolute max. and the absolute min. exist in V. In particular if M is the maximum and m is the minimum, then m ≤ f(x) ≤ M for every x in the interval [a, b]. So a continuous function defined over a closed interval V is always bounded.
  • 30.
    Continuous and DifferentiableFunctions The other important fact about continuous functions is the existence of extrema over a closed interval. II. Extrema Theorem for Continuous Functions Let y = f(x) be a continuous function defined over a closed interval V = [a, b], then both the absolute max. and the absolute min. exist in V. In particular if M is the maximum and m is the minimum, then m ≤ f(x) ≤ M for every x in the interval [a, b]. So a continuous function defined over a closed interval V is always bounded. Corollary. Let y = f(x) be an elementary function defined over a closed interval V = [a, b], then f(x) is continuous over [a, b], hence both the absolute max. and the absolute min. exist in V.
  • 31.
    Continuous and DifferentiableFunctions Differentiable Functions
  • 32.
    Continuous and DifferentiableFunctions Differentiable Functions We state the following important theorems about differentiable functions without proofs.
  • 33.
    Continuous and DifferentiableFunctions Differentiable Functions We state the following important theorems about differentiable functions without proofs. Differentiability is a lot stronger condition than continuity at a point on a graph.
  • 34.
    Continuous and DifferentiableFunctions Differentiable Functions We state the following important theorems about differentiable functions without proofs. Differentiability is a lot stronger condition than continuity at a point on a graph. Theorem. If f(x) is differentiable at x = a then it is continuous at x = a.
  • 35.
    Continuous and DifferentiableFunctions Differentiable Functions We state the following important theorems about differentiable functions without proofs. Differentiability is a lot stronger condition than continuity at a point on a graph. Theorem. If f(x) is differentiable at x = a then it is continuous at x = a. Differentiability means the rate of change may be measured. This observation leads to Rolle’s Theorem which gives the existence of a point c where f'(c) = 0.
  • 36.
    Continuous and DifferentiableFunctions Differentiable Functions We state the following important theorems about differentiable functions without proofs. Differentiability is a lot stronger condition than continuity at a point on a graph. Theorem. If f(x) is differentiable at x = a then it is continuous at x = a. Differentiability means the rate of change may be measured. The rate of change at an extremum x = c of f(x) must be 0 because it can’t be + (increasing) or – (decreasing) hence f'(c) = 0.
  • 37.
    Continuous and DifferentiableFunctions Differentiable Functions We state the following important theorems about differentiable functions without proofs. Differentiability is a lot stronger condition than continuity at a point on a graph. Theorem. If f(x) is differentiable at x = a then it is continuous at x = a. Differentiability means the rate of change may be measured. The rate of change at an extremum x = c of f(x) must be 0 because it can’t be + (increasing) or – (decreasing) hence f'(c) = 0. This observation leads to Rolle’s Theorem which gives the existence of a point c where f'(c) = 0.
  • 38.
    Continuous and DifferentiableFunctions Rolle’s Theorem Let f(x) be a differentiable function defined over the closed interval [a, b] with a < b and that f(a) = f(b),
  • 39.
    Continuous and DifferentiableFunctions Rolle’s Theorem Let f(x) be a differentiable function defined over the closed interval [a, b] with a < b and that f(a) = f(b), f(a)=f(b) x a c b x
  • 40.
    Continuous and DifferentiableFunctions Rolle’s Theorem Let f(x) be a differentiable function defined over the closed interval [a, b] with a < b and that f(a) = f(b), then there is at least one c where a < c < b such that f'(c) = 0. other c’s f'(c) = 0 f(a)=f(b) x a c b x
  • 41.
    Continuous and DifferentiableFunctions Rolle’s Theorem Let f(x) be a differentiable function defined over the closed interval [a, b] with a < b and that f(a) = f(b), then there is at least one c where a < c < b such that f'(c) = 0. other c’s Proof. Consider the following f'(c) = 0 two cases. f(a)=f(b) x a c b x
  • 42.
    Continuous and DifferentiableFunctions Rolle’s Theorem Let f(x) be a differentiable function defined over the closed interval [a, b] with a < b and that f(a) = f(b), then there is at least one c where a < c < b such that f'(c) = 0. other c’s Proof. Consider the following f'(c) = 0 two cases. 1. The function f(x) is a f(a)=f(b) constant function, i.e. x a c b x f(x) = f(a) = k, then f'(x) = 0
  • 43.
    Continuous and DifferentiableFunctions Rolle’s Theorem Let f(x) be a differentiable function defined over the closed interval [a, b] with a < b and that f(a) = f(b), then there is at least one c where a < c < b such that f'(c) = 0. other c’s Proof. Consider the following f'(c) = 0 two cases. 1. The function f(x) is a f(a)=f(b) constant function, i.e. x a c b x f(x) = f(a) = k, then f'(x) = 0. Any number c where a < c < b would satisfy f'(c) = 0.
  • 44.
    Continuous and DifferentiableFunctions Rolle’s Theorem Let f(x) be a differentiable function defined over the closed interval [a, b] with a < b and that f(a) = f(b), then there is at least one c where a < c < b such that f'(c) = 0. other c’s Proof. Consider the following f'(c) = 0 two cases. 1. The function f(x) is a f(a)=f(b) constant function, i.e. x a c b x f(x) = f(a) = k, then f'(x) = 0 Any number c where a < c < b would satisfy f'(c) = 0. 2. If the function f(x) is not a constant function, then there exists an extremum c between a and b, with f(c) ≠ f(a) and f(c) ≠ (b) and we must have f'(c) = 0.
  • 45.
    Continuous and DifferentiableFunctions The average rate of change of y = f(x) from x = a to b is f(b) – f(a) = Δy = slope of the chord as shown. Δx b–a Avg. rate of change y = f(x) = slope of the chord (b, f(b)) Δy = Δx Δy Δx (a, f(a)) a b
  • 46.
    Continuous and DifferentiableFunctions The average rate of change of y = f(x) from x = a to b is f(b) – f(a) = Δy = slope of the chord as shown. Δx b–a The graph y = f(x) is the rotation of the graph of some function y = g(x) defined over some interval [A, B] Avg. rate of change y = g(x) y = f(x) = slope of the chord (b, f(b)) g(A) = g(B) Δy = Δx (A, g(A)) (B, g(B)) Δy Δx (a, f(a)) Rotate A C B a b
  • 47.
    Continuous and DifferentiableFunctions The average rate of change of y = f(x) from x = a to b is f(b) – f(a) = Δy = slope of the chord as shown. Δx b–a The graph y = f(x) is the rotation of the graph of some function y = g(x) defined over some interval [A, B] with g(A) = g(B) with the rotation taking (A,g(A)) to (a, f(a)) and (B,g(B)) to (b, f(b)) as shown. Avg. rate of change y = g(x) y = f(x) = slope of the chord (b, f(b)) g(A) = g(B) Δy = Δx (A, g(A)) (B, g(B)) Δy Δx (a, f(a)) Rotate A C B a b
  • 48.
    Continuous and DifferentiableFunctions If in addition y = g(x) is differentiable over an interval that contains [A, B], then Rolle’s Theorem implies the existence of at least one C where A < C < B such that the tangent line at (C, g(C)) is a horizontal. y = f(x) Rolle’s Theorem y = g(x) (b, f(b)) g(A) = g(B) (A, g(A)) (B, g(B)) (a, f(a)) There exists a C where g'(C) = 0 Rotate A C B a b
  • 49.
    Continuous and DifferentiableFunctions If in addition y = g(x) is differentiable over an interval that contains [A, B], then Rolle’s Theorem implies the existence of at least one C where A < C < B such that the tangent line at (C, g(C)) is a horizontal. Under the rotation this horizontal line rotates into a tangent line that is parallel to the chord from (a, f(a)) to (b, f(b)) y = f(x) Rolle’s Theorem y = g(x) (b, f(b)) g(A) = g(B) (A, g(A)) (B, g(B)) (a, f(a)) There exists a C There exists a C where where g'(C) = 0 Rotate f '(C) = Avg. rate of change A C B a b
  • 50.
    Continuous and DifferentiableFunctions If in addition y = g(x) is differentiable over an interval that contains [A, B], then Rolle’s Theorem implies the existence of at least one C where A < C < B such that the tangent line at (C, g(C)) is a horizontal. Under the rotation this horizontal line rotates into a tangent line that is parallel to the chord from (a, f(a)) to (b, f(b)) which gives us the Mean Value Theorem. y = f(x) Rolle’s Theorem Mean Value Theorem y = g(x) (b, f(b)) g(A) = g(B) (A, g(A)) (B, g(B)) (a, f(a)) There exists a C There exists a C where where g'(C) = 0 Rotate f '(C) = Avg. rate of change A C B a b
  • 51.
    Continuous and DifferentiableFunctions Mean Value Theorem Let y = f(x) be a differentiable function over an interval that contains [a, b], then there is at least one c where a < c < b such that y = f(x) f '(c) = f(b) – f(a) . slope = Avg. rate of change (b, f(b)) b–a = f(b) – f(a) b–a . (a, f(a)) a c b There exists a c where f(b) – f(a) f '(c) = Avg. rate of change = b–a .
  • 52.
    Continuous and DifferentiableFunctions Mean Value Theorem Let y = f(x) be a differentiable function over an interval that contains [a, b], then there is at least one c where a < c < b such that y = f(x) f '(c) = f(b) – f(a) . slope = Avg. rate of change (b, f(b)) b–a = f(b) – f(a) b–a . Remarks 1. The precise condition for the theorem is that (a, f(a)) f(x) is continuous in [a, b], and differentiable in (a, b). The condition above is stronger but is sufficient a c b for our purposes. There exists a c where f(b) – f(a) f '(c) = Avg. rate of change = b–a .
  • 53.
    Continuous and DifferentiableFunctions 2. In statistics, the word “Mean” denotes the “average”. However, the statement of, the “Mean Value” in the Mean Value Theorem refers to the average value of the rate–of–change, y = f(x) slope = Avg. rate of change (b, f(b)) f(b) – f(a) = b–a . (a, f(a)) a c b There exists a c where f(b) – f(a) f '(c) = Avg. rate of change = b–a .
  • 54.
    Continuous and DifferentiableFunctions 2. In statistics, the word “Mean” denotes the “average”. However, the statement of, the “Mean Value” in the Mean Value Theorem refers to the average value of the rate–of–change, y = f(x) not the “mean value” slope = Avg. rate of change (b, f(b)) f(b) – f(a) or the average value of = b–a . the function f(x) over the interval [a, b]. (a, f(a)) a c b There exists a c where f(b) – f(a) f '(c) = Avg. rate of change = b–a .
  • 55.
    Continuous and DifferentiableFunctions 2. In statistics, the word “Mean” denotes the “average”. However, the statement of, the “Mean Value” in the Mean Value Theorem refers to the average value of the rate–of–change, y = f(x) not the “mean value” slope = Avg. rate of change (b, f(b)) f(b) – f(a) or the average value of = b–a . the function f(x) over the interval [a, b]. The average value of (a, f(a)) the function f(x) over the interval [a, b] is defined by integrals which are c our next topics. a b There exists a c where f(b) – f(a) f '(c) = Avg. rate of change = b–a .
  • 56.
    Continuous and DifferentiableFunctions Example A. We left Los Angeles at 12:00 PM arrived at San Francisco at 3:30 PM that same afternoon covering a distance of 350 miles.
  • 57.
    Continuous and DifferentiableFunctions Example A. We left Los Angeles at 12:00 PM arrived at San Francisco at 3:30 PM that same afternoon covering a distance of 350 miles. Therefore our average rate is 350 / 3½ miles/hr = 100 mph.
  • 58.
    Continuous and DifferentiableFunctions Example A. We left Los Angeles at 12:00 PM arrived at San Francisco at 3:30 PM that same afternoon covering a distance of 350 miles. Therefore our average rate is 350 / 3½ miles/hr = 100 mph. By the Mean Value Theorem, our speedometer must display the speed at exactly 100mph at some point in time during our trip.
  • 59.
    Continuous and DifferentiableFunctions Example A. We left Los Angeles at 12:00 PM arrived at San Francisco at 3:30 PM that same afternoon covering a distance of 350 miles. Therefore our average rate is 350 / 3½ miles/hr = 100 mph. By the Mean Value Theorem, our speedometer must display the speed at exactly 100mph at some point in time during our trip. Example B. Find the x and y intercepts of the line that is tangent to the graph y = √2x + 1 and is parallel to the chord connecting the points with x = 0 and x = 4. Use a graphing software to draw the graph to confirm your answers.
  • 60.
    Continuous and DifferentiableFunctions Example A. We left Los Angeles at 12:00 PM arrived at San Francisco at 3:30 PM that same afternoon covering a distance of 350 miles. Therefore our average rate is 350 / 3½ miles/hr = 100 mph. By the Mean Value Theorem, our speedometer must display the speed at exactly 100mph at some point in time during our trip. Example B. Find the x and y intercepts of the line that is tangent to the graph y = √2x + 1 and is parallel to the chord connecting the points with x = 0 and x = 4. Use a graphing software to draw the graph to confirm your answers. The end points in question are (0, 1) and (4, 3) so the chord connecting them has slope ½.
  • 61.
    Continuous and DifferentiableFunctions The function y = √2x + 1 is differentiable and satisfies the condition of the Mean Value Theorem, therefore there is some x = c where 0 < c < 4 and y'(c) = ½ .
  • 62.
    Continuous and DifferentiableFunctions The function y = √2x + 1 is differentiable and satisfies the condition of the Mean Value Theorem, therefore there is some x = c where 0 < c < 4 and y'(c) = ½ . We may solve for c directly.
  • 63.
    Continuous and DifferentiableFunctions The function y = √2x + 1 is differentiable and satisfies the condition of the Mean Value Theorem, therefore there is some x = c where 0 < c < 4 and y'(c) = ½ . We may solve for c directly. y = √2x + 1 → y'(x) = 1/√2x + 1
  • 64.
    Continuous and DifferentiableFunctions The function y = √2x + 1 is differentiable and satisfies the condition of the Mean Value Theorem, therefore there is some x = c where 0 < c < 4 and y'(c) = ½ . We may solve for c directly. y = √2x + 1 → y'(x) = 1/√2x + 1 Set 1 = 1 √2x + 1 2
  • 65.
    Continuous and DifferentiableFunctions The function y = √2x + 1 is differentiable and satisfies the condition of the Mean Value Theorem, therefore there is some x = c where 0 < c < 4 and y'(c) = ½ . We may solve for c directly. y = √2x + 1 → y'(x) = 1/√2x + 1 Set 1 = 1 √2x + 1 2 √2x + 1 = 2
  • 66.
    Continuous and DifferentiableFunctions The function y = √2x + 1 is differentiable and satisfies the condition of the Mean Value Theorem, therefore there is some x = c where 0 < c < 4 and y'(c) = ½ . We may solve for c directly. y = √2x + 1 → y'(x) = 1/√2x + 1 Set 1 = 1 √2x + 1 2 √2x + 1 = 2 2x + 1 = 4 x = 3/2
  • 67.
    Continuous and DifferentiableFunctions The function y = √2x + 1 is differentiable and satisfies the condition of the Mean Value Theorem, therefore there is some x = c where 0 < c < 4 and y'(c) = ½ . We may solve for c directly. y = √2x + 1 → y'(x) = 1/√2x + 1 Set 1 = 1 √2x + 1 2 √2x + 1 = 2 2x + 1 = 4 x = 3/2 Therefore the tangent line passes through (3/2, 2) and it has slope ½.
  • 68.
    Continuous and DifferentiableFunctions The function y = √2x + 1 is differentiable and satisfies the condition of the Mean Value Theorem, therefore there is some x = c where 0 < c < 4 and y'(c) = ½ . We may solve for c directly. y = √2x + 1 → y'(x) = 1/√2x + 1 Set 1 = 1 √2x + 1 2 √2x + 1 = 2 2x + 1 = 4 x = 3/2 Therefore the tangent line passes through (3/2, 2) and it has slope ½. So the equation of the tangent line in question is y = ½ (x – 3/2) + 2 or y = x/2 + 5/4. Its x intercept is at –5/2 and the y intercept is at 5/4.