Memory Review Chapter 5 Calc BC

Essential formulas and information for Chapter 5

d/dx(sin(x))

d/dx(sin(x))

cos(x)

d/dx(cos(x))

-sin(x)

d/dx(tan(x))

sec²x

d/dx(csc(x))

-cscxcotx

d/dx(sec(x))

secxtanx

d/dx(cot(x))

-csc²x

d/dx(arcsin(x))

(1)/(√1-x²)

d/dx(arccos(x))

(-1)/(√1-x²)

d/dx(arctan(x))

(1)/(1+x²)

d/dx(arccsc(x))

(-1)/(|x|√x²-1)

d/dx(arcsec(x))

(1)/(|x|√x²-1)

d/dx(arccot(x))

(-1)/(1+x²)

d/dx(e^x)

e^x

d/dx(a^x)

(a^x)(ln(a))

d/dx(ln(x))

1/x

d/dx(log_a(x))

(1/x)(1/ln(a))

d/dx(f(x)g(x))

(f’(x)g(x))+(f(x)g’(x))

d/dx(f(x)/g(x))

((g(x)f’(x))-(f(x)g’(x)))/(g(x))²

d/dx(f(g(x)))

f’(g(x))*g’(x)

g’(x) < 0 means

g(x) is decreasing

g’(x) > 0 means

g(x) is increasing

g”(x) < 0 means

g(x) is concave down

g”(x) > 0 means

g(x) is concave up

g’(x) changed from + to - means

g(x) has a relative maximum

g’(x) changed from - to + means

g(x) has a relative minimum

g”(x) changes signs means

g(x) has a point of inflection

g’(x) = 0 and g”(x) is + means

g(x) has a minimum

g’(x) = 0 and g”(x) is - means

g(x)has a maximum

critical point

interior point on domain of g(x) such that g’(x) = 0 or is undefined

Candidates test

Locate all critical points and endpoints. Test g(x) to find maximums and minimums.

MVT Conditions

f(x) is continuous in the closed interval [a,b] and is differentiable over (a,b)

MVT Guarantee

c in (a,b)

f’(C) = (f(b)-f(a))/(b-a)

Rolle’s Theorem Conditions

f(x) is continuous over [a,b] and differentiable over (a,b) and f(a) = f(b)

Rolle’s Theorem Guarentee

c in (a,b)

f’(C) = 0

IVT Conditions

f(x) is continuous over [a,b] and f(a)<M<f(b)

IVT Guarentee

f(x) takes on every value between f(a) and f(b), then f(C) = M for some c

EVT Conditions

f(x) is continuous over [a,b]

EVT Guarentee

c in [a,b]

f(C)≥ f(x) for all x in [a,b]

f(C)≤f(x) for all x