Flashcards
Calculus - Chapter Three
Cards in this deck (28)
- 1LIMITS a) Let k element of R and let f:R tends to R be the function given by f(x)=k, then for any c element of R, f(x) tends to ... b) Let f:R tends to R be the function given by f(x)=x, then for any c element of R, f(x) tends to ...a) f(x) tends to k as x tends to c
b) f(x) tends to c as x tends to c - 2c) For any c element of R, modulus(x) tends to ... d) If f(x) tends to l and x tends to c and f(x) tends to m as x tends to c then ... e) If f(x)>/0 for all x element I/{c} and f(x) tends to l as x tends to c then ... f) State the Sandwich Theorem for functionsc) modulus(x) tends to modulus(c) as x tends to c
d) l=m
e) l>/0
f) If g(x) tends to l and h(x) tends to l as x tends to c and if g(x) - 3a) Suppose f is defined on an open interval I, then f(x) tends to l as x tends to c if and only if ... b) Suppose f is defined on an open interval I, then f(x) tends to f(c) as x tends to c if and only if ...a) ... for every sequence {xn} converging to c, f(xn) tends to l as n tends to infinity
b) ... for every sequence {xn} converging to c, f(xn) tends to f(c) as n tends to infinity - 4a) If there is a sequence {xn} that converges to c but f(xn) doesn't tend to l as n tends to infinity then ... b) If there are sequences {un} and {vn} converging to c such that {f(un)} and {f(vn)} converge to different limits then ...a) ... f(x) doesn't tend to l as x tends to c
b) ... f has no limits as x tends to c - 5PROPERTIES OF LIMITS Let I be an open interval in R and let c element I. Let f,g, be functions defined on I such that f(x) tends to l and g(x) tends to m as x tends to c. Then a) f(x) = g(x) tends to ... b) f(x)g(x) tends to ... c) If k element of R, then kf(x) tends to ...a) f(x) + g(x) tends to l + m as x tends to c
b) f(x)g(x) tends to lm as x tends to c
c) kf(x) tends to kl as x tends to c - 6d) If g is nonzero on I and m doesn't =0 then f(x)/g(x) tends to ...d) l/m as x tends to c
- 7For any polynomial p and any c element of R: lim (as x tends to c) [p(x)] = ?lim (as x tends to c)[p(x)] = p(c)
- 8CONTINUITY Let f be defined on an open interval I and let c element of I. We say that f is continuous at c if ... and discontinuous at c if ... Let f be defined on an open interval I. We say that f is continuous on I if ... and discontinuous on I ...... lim (as x tends to c) [f(x)] = f(c)
... otherwise
... f is continuous at every point of I ... otherwise - 9On R every polynomial function is ...... continuous
- 10Suppose a function f is defined on an open interval (a,b) (and possibly at a and/or b). W say that: a) f(x) tends to l and x tends to a^+ if ... b) f(x) tends to m as x tends to b^- if ...a) ... for every epsilon>0 there exists a delta>0 such that: If x element (a,b) and 00 there exists a delta>0 such that: If x element (a,b) and 0
- 11Let f be defined on a closed interval [a,b[. We say that f is continuous on [a,b] if ...... f is continuous at every point of (a,b), f(x) tends to f(a) as x tends to a^+ and f(x) tends to f(b) as x tends to b^-
- 12a) Let the real valued function f be defined on an open interval I, except possibly at a point c element of I. Then f(x) tends to l as x tends to c if and only if ...a) ... f(x) tends to l as x tends to c^- and f(x) tends to l as x tends to c^+
- 13b) Suppose a real valued function is continuous on an open interval I and a,b, element I with ab) ... restriction to [a,b] and also on restriction to (a,b)
- 14State the six standard functions that are continuous and the domains on which this occurssin: R tends to R
cos: R tends to R
exp: R tends to R
ln: (0,infinity) tends to R
(sqrt)^n: R^+ tends to R (n element of N)
tan: R\{pi/2+mpi: m element of Z} tends to R - 15State the Algebra of Continuous FunctionsSuppose that funtions f,g,: I tends to R are continuous at c element of I and that k element of R (where I is an interval).
The f+g,kf,fg are continuous at c, and if g(c) doesn't equal 0 then f/g is continuous at c.
If f,g are continuous on I then so are f+g, kf,fg and (if g doesn't =0) f/g. - 16A rational function is a function of the form ... A rational function is continuous at ...... f:R\{x:q(x)=0} tends to R, where f(x) = p(x)/q(x) = and p,q are polynomials
... every point of its domain - 17State the Chain Rule for Continuous Functions PROOF ON SEPARATE SHEETSuppose that I1, I2, I3 are open intervals in R and let c element of I1.
Suppose further that f: I1 tends to I2 is continuous at c and that g: I2 tends to I3 is continuous at f(c).
Then g dot f : I1 tends to I3 is continuous at c - 18State the Intermediate Value TheoremLet f:[a,b] tends to R be continuous on [a,b] and suppose that either f(a)gamma>f(b). Then there exists c element (a,b) such that f(c) = gamma
- 19DIFFERENTIABILITY The gradient of the line joining (a,f(a)) and (x,f(x)) is ... f:I tends to R is differentiable at a element I if ...... [f(x)-f(a)]/(x-a)
... lim (as x tends to a) [[f(x)-f(a)]/(x-a)] exists. - 20If the limit exists it is called what?The derivative of f at a, denoted f'(a)
- 21If f is differentiable at a then ... PROOF ON SEPARATE SHEET!!... f is continuous at a
- 22If a function f is not continuous at a then ... Continuous functions however don't need to be ...... it cannot be differentiable
... differentiable - 23Algebra of Differentiable Functions: Let I be an open interval. If f,g:I tends to R are differentiable at a element of I then so are: a) f+g with ... b) fg with ... c) [if g(a) =/0] f/g with ...a) (f+g)'(a) = f'(a) + g'(a)
b) (fg)'(a) = f'(a)g(a) + f(a)g'(a)
c) (f/g)'(a) = [f'(a)g(a) - f(a)g'(a)]/g(a)^2 - 24The Chain Rule: If g is differentiable at a and f is differentiable at g(a) then ...... f dot g is differentiable at a, and
(f dot g)'(x) = (f' dot g)(x) times g'(x) - 25Polynomials are ... Rational functions are ...... differentiable
... differentiable where they are defined - 26Suppose that I is an open interval and that f:I tends to R. We say that the function f is differentiable on I if ...... it is differentiable at every point of I. This includes the possibility that I = R.
- 27Rolle's Theorem: Suppose a,b element of R with a... x element (a,b) such that f'(c) = 0
- 28State the Mean Value Theorem PROOF ON SEPARATE SHEET!!Let f:[a,b] tends to R be continuous on [a,b] and differentiable on (a,b).
Then there exists c element (a,b) such that
f'(c) = [f(b)-f(a)]/(b-a)
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- Flashcards on Calculus - Chapter Three, created by Cara Wales on 18/05/2016.
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