MATH 104 FINAL EXAM WITH
ANSWERS
Taylor Polynomial Formula - .......ANSWERS 🔷🖊✔✔For the nth degree Taylor
polynomial, take the (n-1)th degree polynomial and add the following term at the
end.
Taylor Polynomial Error - .......ANSWERS 🔷🖊✔✔|f(x)-Tn(x)| = |f^(n+1)(c)(x-
a)^(n+1)/(n+1)!
coefficient ck - .......ANSWERS 🔷🖊✔✔ck = f^k(a)/k!
Tn^(n)(a) - .......ANSWERS 🔷🖊✔✔= n!c_n
Mean Value Theorem - .......ANSWERS 🔷🖊✔✔Let a and b be real numbers, a<b. Let
f(x) be:
- cont' over [a, b]
- diff over (a, b)
There is a c within (a, b) such that
f'(c) = f(b)-f(a)/b-a
Rolle's Th'm: f(b) = f(ab) -> then there is a c such that f'(c) = 0.
Global Max/Min - .......ANSWERS 🔷🖊✔✔INCLUDES ENDPOINTS, + CP and SP.
Endpoints can be CP/SP, can't be local extrema.
Local Max/Min - .......ANSWERS 🔷🖊✔✔Take first derivative and find CP/SP.
Curve Sketching - .......ANSWERS 🔷🖊✔✔- domain, where x DNE
- intercepts, f(x) = 0 and x = 0
- vertical asymptotes - find LH and RH limits at these values + SYMMETRY
, - horizontal asymptotes - evaluate limits of your fx as x-> +/- infinity. (When limit
evaluates to +/- infinity, no HA exist). BOBO BOTN EATS DC. Test if f(x) crosses HA -
> f(x) = HA
First Derivative Test - .......ANSWERS 🔷🖊✔✔Find intervals over which fx is
increasing or decreasing (f'(x)>0 or f'(x)<0)
- find CP and SP
Second Derivative Test - .......ANSWERS 🔷🖊✔✔f''(x)>0, concave up. f''(x)<0,
concave down. f''(x) changes sign - inflection point.
Vertical tangent lines occur when - .......ANSWERS 🔷🖊✔✔f/(x) -> +/- infinity as x->
x0. CAN BE A POINT OF INFLECTION.
inflection point - .......ANSWERS 🔷🖊✔✔For a function f(x), we say x = x0 is an IP if
1) there is a tangent line to v = f(x) at x=x0.
2) concavity changes as we pass through (x0, f(x0)). DOES NOT INCLUDE
CUSPS/CORNERS.
Continuity - .......ANSWERS 🔷🖊✔✔f(x) is cont' at x=a when:
a) f(a) is defined
2) lim x->a f(x) exists (RH lim = LH lim_
3) lim x->af(x) = f(a)
Polynomial, exponential, sqrt, log(x), sin and cos
Intermediate Value Theorem - .......ANSWERS 🔷🖊✔✔If f is a cont' fx whose domain
is in [a, b], then there exists at least one c within [a,b] such that f(x) < f(c) < f(b)
If f(a)<0 and f(b)>0, there is at least one solution/root (f(c)=0)
Derivative - .......ANSWERS 🔷🖊✔✔f'(x) = lim h-> 0 [f(x+h)-f(x)/h]
or f'(a).= limx->a [f(x)-f(a)/(x-a)]
ANSWERS
Taylor Polynomial Formula - .......ANSWERS 🔷🖊✔✔For the nth degree Taylor
polynomial, take the (n-1)th degree polynomial and add the following term at the
end.
Taylor Polynomial Error - .......ANSWERS 🔷🖊✔✔|f(x)-Tn(x)| = |f^(n+1)(c)(x-
a)^(n+1)/(n+1)!
coefficient ck - .......ANSWERS 🔷🖊✔✔ck = f^k(a)/k!
Tn^(n)(a) - .......ANSWERS 🔷🖊✔✔= n!c_n
Mean Value Theorem - .......ANSWERS 🔷🖊✔✔Let a and b be real numbers, a<b. Let
f(x) be:
- cont' over [a, b]
- diff over (a, b)
There is a c within (a, b) such that
f'(c) = f(b)-f(a)/b-a
Rolle's Th'm: f(b) = f(ab) -> then there is a c such that f'(c) = 0.
Global Max/Min - .......ANSWERS 🔷🖊✔✔INCLUDES ENDPOINTS, + CP and SP.
Endpoints can be CP/SP, can't be local extrema.
Local Max/Min - .......ANSWERS 🔷🖊✔✔Take first derivative and find CP/SP.
Curve Sketching - .......ANSWERS 🔷🖊✔✔- domain, where x DNE
- intercepts, f(x) = 0 and x = 0
- vertical asymptotes - find LH and RH limits at these values + SYMMETRY
, - horizontal asymptotes - evaluate limits of your fx as x-> +/- infinity. (When limit
evaluates to +/- infinity, no HA exist). BOBO BOTN EATS DC. Test if f(x) crosses HA -
> f(x) = HA
First Derivative Test - .......ANSWERS 🔷🖊✔✔Find intervals over which fx is
increasing or decreasing (f'(x)>0 or f'(x)<0)
- find CP and SP
Second Derivative Test - .......ANSWERS 🔷🖊✔✔f''(x)>0, concave up. f''(x)<0,
concave down. f''(x) changes sign - inflection point.
Vertical tangent lines occur when - .......ANSWERS 🔷🖊✔✔f/(x) -> +/- infinity as x->
x0. CAN BE A POINT OF INFLECTION.
inflection point - .......ANSWERS 🔷🖊✔✔For a function f(x), we say x = x0 is an IP if
1) there is a tangent line to v = f(x) at x=x0.
2) concavity changes as we pass through (x0, f(x0)). DOES NOT INCLUDE
CUSPS/CORNERS.
Continuity - .......ANSWERS 🔷🖊✔✔f(x) is cont' at x=a when:
a) f(a) is defined
2) lim x->a f(x) exists (RH lim = LH lim_
3) lim x->af(x) = f(a)
Polynomial, exponential, sqrt, log(x), sin and cos
Intermediate Value Theorem - .......ANSWERS 🔷🖊✔✔If f is a cont' fx whose domain
is in [a, b], then there exists at least one c within [a,b] such that f(x) < f(c) < f(b)
If f(a)<0 and f(b)>0, there is at least one solution/root (f(c)=0)
Derivative - .......ANSWERS 🔷🖊✔✔f'(x) = lim h-> 0 [f(x+h)-f(x)/h]
or f'(a).= limx->a [f(x)-f(a)/(x-a)]
