f(x)
... 13) Is f defined at x = 2 ? Is f continuous at x = 2? 14) At what values of x is f continuous? 15) What value should be assigned to f(2) to make the extended function continuous at x = 2 . 16) What new value should be assigned to f(1) to make the extended function continuous at x = 1 . 17) Is it pos ...
... 13) Is f defined at x = 2 ? Is f continuous at x = 2? 14) At what values of x is f continuous? 15) What value should be assigned to f(2) to make the extended function continuous at x = 2 . 16) What new value should be assigned to f(1) to make the extended function continuous at x = 1 . 17) Is it pos ...
AP Calculus BC Syllabus - Fort Thomas Independent Schools
... the local linear approximation of the function The definite integral should be understood as the limit of a Riemann sum and as the net accumulation of the rate of change The relationship between derivatives and the definite integral should be understood in terms of both parts of the Fundamental ...
... the local linear approximation of the function The definite integral should be understood as the limit of a Riemann sum and as the net accumulation of the rate of change The relationship between derivatives and the definite integral should be understood in terms of both parts of the Fundamental ...
Lecture notes 2.26.14
... 0 ≤ f (x̄ + λd) − f (x̄) = λ∇f (x̄)T d + o(kλdk) dividing by λ > 0 and letting λ → 0+ , we obtain 0 ≤ ∇f (x̄)T d. Similarly, dividing by λ < 0 and letting λ → 0− , we obtain 0 ≥ ∇f (x̄)T d. As a consequence, f (x̄)T d = 0 for all d ∈ Rn . This implies that ∇f (x̄) = 0. The case where x̄ is a local m ...
... 0 ≤ f (x̄ + λd) − f (x̄) = λ∇f (x̄)T d + o(kλdk) dividing by λ > 0 and letting λ → 0+ , we obtain 0 ≤ ∇f (x̄)T d. Similarly, dividing by λ < 0 and letting λ → 0− , we obtain 0 ≥ ∇f (x̄)T d. As a consequence, f (x̄)T d = 0 for all d ∈ Rn . This implies that ∇f (x̄) = 0. The case where x̄ is a local m ...
A short proof of the Bolzano-Weierstrass Theorem
... Theorem 1 (Bolzano-Weierstrass Theorem, Version 1). Every bounded sequence of real numbers has a convergent subsequence. To mention but two applications, the theorem can be used to show that if [a, b] is a closed, bounded interval and f : [a, b] → R is continous, then f is bounded. One may also invo ...
... Theorem 1 (Bolzano-Weierstrass Theorem, Version 1). Every bounded sequence of real numbers has a convergent subsequence. To mention but two applications, the theorem can be used to show that if [a, b] is a closed, bounded interval and f : [a, b] → R is continous, then f is bounded. One may also invo ...
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.