MAC 2233
... If f ' ' is positive on an open interval I, then f is concave up because slope of tangent is increasing. If f ' ' is negative on an open interval I, then f is concave down because slope of tangent is decreasing. ...
... If f ' ' is positive on an open interval I, then f is concave up because slope of tangent is increasing. If f ' ' is negative on an open interval I, then f is concave down because slope of tangent is decreasing. ...
On Malliavin`s proof of Hörmander`s theorem
... smooth vector bundle of dimension d < n. Then E is integrable if and only if there (locally) exists a smooth foliation of M into leaves of dimension d such that, for every x ∈ M, the tangent space of the leaf passing through x is given by Ex . In view of this S result, Hörmander’s condition is not ...
... smooth vector bundle of dimension d < n. Then E is integrable if and only if there (locally) exists a smooth foliation of M into leaves of dimension d such that, for every x ∈ M, the tangent space of the leaf passing through x is given by Ex . In view of this S result, Hörmander’s condition is not ...
Have a great summer! Ponder!
... e. Compound Functions (chain rule) 2. Explaining what is meant by a derivative (What does the new derivative function tell you both in mathematical terms and in physics terms. For example, the derivative of a displacement function gives you a new function that calculates the slope of the displacemen ...
... e. Compound Functions (chain rule) 2. Explaining what is meant by a derivative (What does the new derivative function tell you both in mathematical terms and in physics terms. For example, the derivative of a displacement function gives you a new function that calculates the slope of the displacemen ...
6 Continuous functions
... to prove that f is continuous at an arbitrary real number a. Example 6.1.5. Let f (x) = x for all x ∈ R. Use Definition 6.1.2 to prove that f is continuous at an arbitrary real number a. Example 6.1.6. Use ǫ-δ definition of continuity, that is Definition 6.1.2, to prove that the function f (x) = 1/x ...
... to prove that f is continuous at an arbitrary real number a. Example 6.1.5. Let f (x) = x for all x ∈ R. Use Definition 6.1.2 to prove that f is continuous at an arbitrary real number a. Example 6.1.6. Use ǫ-δ definition of continuity, that is Definition 6.1.2, to prove that the function f (x) = 1/x ...
Fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of the derivative of a function with the concept of the function's integral.The first part of the theorem, sometimes called the first fundamental theorem of calculus, is that the definite integration of a function is related to its antiderivative, and can be reversed by differentiation. This part of the theorem is also important because it guarantees the existence of antiderivatives for continuous functions.The second part of the theorem, sometimes called the second fundamental theorem of calculus, is that the definite integral of a function can be computed by using any one of its infinitely-many antiderivatives. This part of the theorem has key practical applications because it markedly simplifies the computation of definite integrals.