Continuity and Differentiability

Properties of differentiable functions

• A function is said to be differentiable at a point if the derivative of the function exists at that point.

• If a function is differentiable at a point, then it is also continuous at that point.

• Differentiability implies continuity, but continuity does not imply differentiability.

• Differentiability is a stronger condition than continuity.

• If a function is differentiable at a point, then it is also differentiable on any interval that contains that point.

• If a function is differentiable at a point, then it is also continuous at that point.

• If a function is differentiable on an interval, then it is continuous on that interval.

• If a function is differentiable on an interval, then it is also differentiable at every point in the interior of the interval.

• The sum/difference/product/quotient of two differentiable functions is also differentiable on any interval where the functions are both differentiable.

• The composition of two differentiable functions is also differentiable on any interval where the functions are both differentiable.

• The derivative of a constant function is 0.

• The derivative of the identity function is 1.

• The power rule: if f(x) = x^n, where n is a constant, then f’(x) = nx^(n-1).

• The constant multiple rule: if f(x) = c * g(x), where c is a constant, then f’(x) = c * g’(x).

• The sum/difference rule: if f(x) = g(x) ± h(x), then f’(x) = g’(x) ± h’(x).

• The product rule: if f(x) = g(x) * h(x), then f’(x) = g’(x) * h(x) + g(x) * h’(x).

• The quotient rule: if f(x) = g(x) / h(x), then f’(x) = (g’(x) * h(x) - g(x) * h’(x)) / h(x)^2.

11. Continuity and Differentiability - Properties of differentiable functions

• The graph of a differentiable function is smooth and has no sharp corners or vertical tangent lines.
• The derivative of a differentiable function represents the rate of change of the function at a particular point.
• The derivative of a differentiable function can be interpreted as the slope of the tangent line to the graph of the function at that point.

12. The Mean Value Theorem

• The Mean Value Theorem states that if a function f(x) is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a point c in the interval (a, b) such that f’(c) = (f(b) - f(a)) / (b - a).
• Geometrically, this means that there exists at least one point on the graph of the function where the tangent line is parallel to the secant line between the endpoints of the interval.

13. Rolle’s Theorem

• Rolle’s Theorem is a special case of the Mean Value Theorem where f(a) = f(b).
• Rolle’s Theorem states that if a function f(x) is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), and if f(a) = f(b), then there exists a point c in the interval (a, b) such that f’(c) = 0.

14. Increasing and Decreasing Functions

• A function is said to be increasing on an interval if the derivative of the function is positive on that interval.
• A function is said to be decreasing on an interval if the derivative of the function is negative on that interval.
• If the derivative of a function is zero on an interval, then the function may have local maximum or minimum points on that interval.

15. First Derivative Test

• The first derivative test is a method to determine the local maximum and minimum points of a function.
• If the derivative changes from positive to negative at a point, then that point is a local maximum.
• If the derivative changes from negative to positive at a point, then that point is a local minimum.

16. Second Derivative Test

• The second derivative test is a method to determine the concavity and the nature of the points of inflection of a function.
• If the second derivative is positive at a point, then the graph is concave up at that point.
• If the second derivative is negative at a point, then the graph is concave down at that point.
• If the second derivative changes sign at a point, then that point is a point of inflection.

17. Optimization Problems

• Optimization problems involve finding the maximum or minimum values of a function subject to certain constraints.
• The critical points of the function (where the derivative is zero or does not exist) and the endpoints of the interval are the potential maximum or minimum points.
• To find the absolute maximum or minimum, evaluate the function at these points and compare the values.

18. Related Rates

• Related rates problems involve finding the rate at which one quantity changes with respect to another quantity in a given situation.
• To solve related rates problems, use the chain rule to take the derivative of the equation that relates the two quantities.
• Substitute the given values and solve for the desired rate of change.

19. Implicit Differentiation

• Implicit differentiation is used to find the derivative of a function that is not given explicitly in the form y = f(x).
• To implicitly differentiate a function, treat y as a function of x, differentiate both sides of the equation with respect to x, and solve for y'.

20. Linear Approximation

• Linear approximation is a method to estimate the value of a function near a known point on the graph.
• The linear approximation of the function f(x) near the point (a, f(a)) is given by the equation L(x) = f(a) + f’(a)(x - a).
• The linear approximation can be used to approximate the value of the function and to find the tangent line to the graph at the point.

21. Continuity and Differentiability - Properties of differentiable functions

• The intermediate value property: If a function f(x) is continuous on the closed interval [a, b], and if k is any number between f(a) and f(b), then there exists at least one point c in the interval (a, b) such that f(c) = k.
• The extreme value theorem: If a function f(x) is continuous on a closed interval [a, b], then f(x) has both a maximum and minimum value on that interval.
• The Darboux property: If a function f(x) is differentiable on an interval, then the derivative of the function also satisfies the intermediate value property.
• The local linearity property: If a function f(x) is differentiable at a point, then the function can be approximated by a linear function near that point.

22. Mean Value Theorem - Example

• Example:
  • Let f(x) = x^2 on the interval [1, 3].
  • The function f(x) is continuous and differentiable on the open interval (1, 3).
  • Using the Mean Value Theorem, there exists a point c in the interval (1, 3) such that f’(c) = (f(3) - f(1))/(3 - 1).
  • Evaluating f’(x) = 2x, we get 2c = (9 - 1)/(2).
  • Solving for c, we find c = 5/2.

23. Rolle’s Theorem - Example

• Example:
  • Let f(x) = x^3 - 4x^2 + 3x - 2.
  • The function f(x) is continuous on the interval [0, 3] and differentiable on the open interval (0, 3).
  • By checking the endpoints, we find that f(0) = -2 and f(3) = 8.
  • Since f(0) = f(3), we can apply Rolle’s Theorem, which states that there exists a point c in the interval (0, 3) such that f’(c) = 0.
  • To find the value of c, we differentiate f(x) to get f’(x) = 3x^2 - 8x + 3.
  • Solving f’(x) = 0, we find two solutions: x = 1 and x = 1/3.

24. Increasing and Decreasing Functions - Example

• Example:
  • Let f(x) = x^3 - 6x^2 + 9x + 1.
  • To determine where the function is increasing or decreasing, we need to find the intervals where f’(x) > 0 and f’(x) < 0.
  • Differentiating f(x) to get f’(x) = 3x^2 - 12x + 9.
  • Setting f’(x) > 0, we have 3x^2 - 12x + 9 > 0.
  • Factoring, we get (x - 1)(x - 3) > 0.
  • The interval where f’(x) > 0 is (1, 3).
  • Setting f’(x) < 0, we have 3x^2 - 12x + 9 < 0.
  • Factoring, we get (x - 1)(x - 3) < 0.
  • The interval where f’(x) < 0 is (0, 1) ∪ (3, ∞).

25. First Derivative Test - Example

• Example:
  • Let f(x) = x^3 - 6x^2 + 9x + 1.
  • To determine the local maximum and minimum points of the function, we need to analyze the sign changes of f’(x).
  • The function f’(x) = 3x^2 - 12x + 9 has sign changes at x = 1 and x = 3.
  • By checking the intervals (0, 1) and (3, ∞), we find that f’(x) > 0 for x in (1, 3) and f’(x) < 0 for x in (0, 1) ∪ (3, ∞).
  • Therefore, the function has a local minimum at x = 1 and a local maximum at x = 3.

26. Second Derivative Test - Example

• Example:
  • Let f(x) = x^3 - 6x^2 + 9x + 1.
  • To determine the concavity and the nature of the points of inflection, we need to analyze the sign changes of f’’(x).
  • Differentiating f(x) twice, we get f’’(x) = 6x - 12.
  • Solving f’’(x) = 0, we find x = 2.
  • By checking the intervals (-∞, 2) and (2, ∞), we find that f’’(x) > 0 for x in (2, ∞) and f’’(x) < 0 for x in (-∞, 2).
  • Therefore, the function is concave up on (2, ∞) and concave down on (-∞, 2).
  • The point of inflection is at x = 2.

27. Optimization Problems - Example

• Example:
  • A farmer wants to enclose a rectangular garden with a fence on three sides, using a building as the fourth side.
  • The farmer has 20 meters of fencing material.
  • Let x be the length of the garden and y be the width of the garden.
  • The perimeter of the garden is 2x + y, and it must be equal to 20.
  • The area of the garden is xy, which we want to maximize.
  • We can solve the equation 2x + y = 20 for y to get y = 20 - 2x.
  • Substituting this expression for y in the area equation, we have A = x(20 - 2x).
  • Taking the derivative of A with respect to x, we get A’ = 20 - 4x.
  • Setting A’ = 0, we find x = 5.
  • Therefore, the length of the garden should be 5 meters and the width should be 10 meters for a maximum area of 50 square meters.

28. Related Rates - Example

• Example:
  • A balloon is being inflated at a constant rate of 2 cm^3/s.
  • The balloon has a spherical shape, and its radius is increasing at a rate of 0.4 cm/s.
  • We want to find the rate at which the volume of the balloon is increasing when the radius is 5 cm.
  • The volume of a sphere is given by the formula V = (4/3)πr^3.
  • Taking the derivative of V with respect to t, we get dV/dt = 4πr^2(dr/dt) by using the chain rule.
  • Substituting the given values, we have dV/dt = 4π(5^2)(0.4) = 8π cm^3/s.
  • Therefore, the volume of the balloon is increasing at a rate of 8π cm^3/s when the radius is 5 cm.

29. Implicit Differentiation - Example

• Example:
  • Consider the equation x^2 + y^2 = 25.
  • To find dy/dx, we need to differentiate both sides of the equation with respect to x.
  • Taking the derivative of x^2 + y^2 = 25, we get 2x + 2y(dy/dx) = 0.
  • Rearranging the equation, we have dy/dx = -2x/2y = -x/y.
  • Hence, the derivative of y with respect to x is given by -x/y.

30. Linear Approximation - Example

• Example:
  • Approximate the value of √16.1 using linear approximation.
  • Choose a known point on the graph, such as f(16) = 4, and its corresponding x-value x = 16.
  • Find the derivative of the function f(x) = √x, which is f’(x) = 1/(2√x).
  • Use the linear approximation equation L(x) = f(x) + f’(x)(x - a).
  • Substituting the values, we have L(16.1) = 4 + (1/(2√16))(16.1 - 16).
  • Evaluating the expression, we get L(16.1) ≈ 4 + (1/8)(0.1) ≈ 4.0125.
  • Therefore, √16.1 is approximately equal to 4.0125.