CALCULUS FINAL EXAM REVIEW
QUESTIONS WITH COMPLETE
ANSWERS
(f + g)' - ANSWER-f' + g'
(fg)' - ANSWER-f'g + fg'
(f /g)' = - ANSWER-(f'g-fg')/g^2
chain rule - ANSWER-f'(g(x))g'(x)
x^a - ANSWER-ax^a-1
e^x - ANSWER-e^x
b^x - ANSWER- b^xlnb
ln x - ANSWER-1/x
logbx - ANSWER-1/xlnb
sin (x) - ANSWER-cos x
cos (x) - ANSWER--sin x
tan (x) - ANSWER-sec^2 x
cot (x) - ANSWER--csc^2 x
dy/dx (x) derivative inverse - ANSWER-1 /(dx/dy)(y)
d/dx (sin^-1 (x)) - ANSWER-1/√(1-x^2)
cos-1 (x) - ANSWER-1/√(1-x^2)
tan^-1 (x) - ANSWER-1/(1+x^2)
cot^-1 (x) - ANSWER--1 /1 + x^2
Implicit Differentiation - ANSWER-take derivative w/ respect to x, solve for dy/dx
log differentiation - ANSWER-log y = v(x)log u(x)
, function extrema - ANSWER-Critical points. Local min (f'' > 0). Local max (f'' < 0).
inflection points - ANSWER-The points at which the curve changes from curving upward
to curving downward (second derivative)
linear approximation - ANSWER-L(x) = f(a) + f'(a)(x-a)
related rates steps - ANSWER-1) Define notation. Respect problem notation.
2) Identify independent/dependent variable.
3) Carry out calculus operations, typically: find the critical points of dependent variable.
for plotting a function - ANSWER-find: Root. Critical point. Inflection point. Increasing.
Decreasing. Concave up, down.
L' Hopital's Rule - ANSWER-lim x->c f(x) g(x) = lim x->c f'(x) g'(x) (0/0) indeterminacy
F(x) = - ANSWER-the integral of f(x) dx + C
F'(x) - ANSWER-= f(x)
definite integral - ANSWER-aintegralb f(x) dx
Riemann Sum - ANSWER-lim n -> ∞ Σ i = 1 to n f(xi*) • Δx
area function - ANSWER-Let f be a continuous function, for t≥a. The area function for f
with left endpoint a is A(x)=∫(^x∨a)f(t) dt, where x≥a. The area function gives the net
area of the region bounded by the graph of f and the t-axis on the interval [a,x].
Fundamental Theorem of Calculus - ANSWER-∫ f(x) dx on interval a to b = F(b) - F(a)
u-substitution - ANSWER-a method of intergration in which f(g(x))*g'(x)dx is rewritten as
f(u)du by substituting u=g(x) and du=g'(x)dx
Intermediate Value Theorem - ANSWER-If f is continuous on [a,b] and k is a number
between f(a) and f(b), then there exists at least one number c such that f(c)=k
horizontal asymptote rules - ANSWER-1. if powers are the same, it is that ratio of the
leading coefficients
2. if top power is greater than bottom, no asymp
3. bottom greater than top, ha=0
also should be used lim as x approaches infinity and neg infinity
vertical asymptote rules - ANSWER-simplify, set denominator=0, roots are vertical
asymptote
QUESTIONS WITH COMPLETE
ANSWERS
(f + g)' - ANSWER-f' + g'
(fg)' - ANSWER-f'g + fg'
(f /g)' = - ANSWER-(f'g-fg')/g^2
chain rule - ANSWER-f'(g(x))g'(x)
x^a - ANSWER-ax^a-1
e^x - ANSWER-e^x
b^x - ANSWER- b^xlnb
ln x - ANSWER-1/x
logbx - ANSWER-1/xlnb
sin (x) - ANSWER-cos x
cos (x) - ANSWER--sin x
tan (x) - ANSWER-sec^2 x
cot (x) - ANSWER--csc^2 x
dy/dx (x) derivative inverse - ANSWER-1 /(dx/dy)(y)
d/dx (sin^-1 (x)) - ANSWER-1/√(1-x^2)
cos-1 (x) - ANSWER-1/√(1-x^2)
tan^-1 (x) - ANSWER-1/(1+x^2)
cot^-1 (x) - ANSWER--1 /1 + x^2
Implicit Differentiation - ANSWER-take derivative w/ respect to x, solve for dy/dx
log differentiation - ANSWER-log y = v(x)log u(x)
, function extrema - ANSWER-Critical points. Local min (f'' > 0). Local max (f'' < 0).
inflection points - ANSWER-The points at which the curve changes from curving upward
to curving downward (second derivative)
linear approximation - ANSWER-L(x) = f(a) + f'(a)(x-a)
related rates steps - ANSWER-1) Define notation. Respect problem notation.
2) Identify independent/dependent variable.
3) Carry out calculus operations, typically: find the critical points of dependent variable.
for plotting a function - ANSWER-find: Root. Critical point. Inflection point. Increasing.
Decreasing. Concave up, down.
L' Hopital's Rule - ANSWER-lim x->c f(x) g(x) = lim x->c f'(x) g'(x) (0/0) indeterminacy
F(x) = - ANSWER-the integral of f(x) dx + C
F'(x) - ANSWER-= f(x)
definite integral - ANSWER-aintegralb f(x) dx
Riemann Sum - ANSWER-lim n -> ∞ Σ i = 1 to n f(xi*) • Δx
area function - ANSWER-Let f be a continuous function, for t≥a. The area function for f
with left endpoint a is A(x)=∫(^x∨a)f(t) dt, where x≥a. The area function gives the net
area of the region bounded by the graph of f and the t-axis on the interval [a,x].
Fundamental Theorem of Calculus - ANSWER-∫ f(x) dx on interval a to b = F(b) - F(a)
u-substitution - ANSWER-a method of intergration in which f(g(x))*g'(x)dx is rewritten as
f(u)du by substituting u=g(x) and du=g'(x)dx
Intermediate Value Theorem - ANSWER-If f is continuous on [a,b] and k is a number
between f(a) and f(b), then there exists at least one number c such that f(c)=k
horizontal asymptote rules - ANSWER-1. if powers are the same, it is that ratio of the
leading coefficients
2. if top power is greater than bottom, no asymp
3. bottom greater than top, ha=0
also should be used lim as x approaches infinity and neg infinity
vertical asymptote rules - ANSWER-simplify, set denominator=0, roots are vertical
asymptote
