CALCULUS EXAM #3 QUESTIONS WITH
VERIFIED ANSWERS
first derivative - ANSWER-tells where graph is increasing or decreasing
second derivative - ANSWER-shows concavity of graph
inflection point - ANSWER-if f(x) is continuous at the point and the graph of the function
changes concavity there
volume equation - ANSWER-v=4/3 r^3
related rate problems - ANSWER-treat all variables as functions of time, use implicit
differentiation to produce equalities which can be solved for the desired rates
linear approximation equation - ANSWER-f'(a)(x-a)+f(a)
linear approximation - ANSWER-approximating functions using the equation of a line,
also known as tangent line approxiamtion
equation for dy - ANSWER-f'(x)dx or f'(x)delta x
dy - ANSWER-differential, dependent on both the values of x and dx, gives the change
in the y value on the tangent line as we move from x to x+dx
dx - ANSWER-equals delta x
equation for delta y - ANSWER-f(x + delta x) - f(x)
absolute max/min - ANSWER-must be a real number which is an actual value and not a
limit of the function, cannot be equal to infinity
critical number - ANSWER-a number "c" in the domain of f such that f'(c)=0 of f'(c) DNE
critical number theorem - ANSWER-if f(x) has a local max or min value at c then c is a
critical number of the function, NOT all critical numbers are local max/min
extreme value theorem - ANSWER-if f(x) is continuous on some closed interval [a,b],
then the function attains an absolute max value f(c) and an absolute min value f(d) at
some points c and d in [a,b]
continuity is key - ANSWER-as long as a function is continuous on a closed interval,
both absolute extrema will occur
VERIFIED ANSWERS
first derivative - ANSWER-tells where graph is increasing or decreasing
second derivative - ANSWER-shows concavity of graph
inflection point - ANSWER-if f(x) is continuous at the point and the graph of the function
changes concavity there
volume equation - ANSWER-v=4/3 r^3
related rate problems - ANSWER-treat all variables as functions of time, use implicit
differentiation to produce equalities which can be solved for the desired rates
linear approximation equation - ANSWER-f'(a)(x-a)+f(a)
linear approximation - ANSWER-approximating functions using the equation of a line,
also known as tangent line approxiamtion
equation for dy - ANSWER-f'(x)dx or f'(x)delta x
dy - ANSWER-differential, dependent on both the values of x and dx, gives the change
in the y value on the tangent line as we move from x to x+dx
dx - ANSWER-equals delta x
equation for delta y - ANSWER-f(x + delta x) - f(x)
absolute max/min - ANSWER-must be a real number which is an actual value and not a
limit of the function, cannot be equal to infinity
critical number - ANSWER-a number "c" in the domain of f such that f'(c)=0 of f'(c) DNE
critical number theorem - ANSWER-if f(x) has a local max or min value at c then c is a
critical number of the function, NOT all critical numbers are local max/min
extreme value theorem - ANSWER-if f(x) is continuous on some closed interval [a,b],
then the function attains an absolute max value f(c) and an absolute min value f(d) at
some points c and d in [a,b]
continuity is key - ANSWER-as long as a function is continuous on a closed interval,
both absolute extrema will occur
