Derivatives
Derivative of:
sin(x), cos(x), tan(x), cot(x), sec(x), cosec(x)
cos(x), -sin(x), sec2(x), -cosec2(x), sec(x)tan(x), -cosec(x)cotan(x)
What is the difference between Rolle's Theorem and MVT?
Rolle's Theorem is a specific case of MVT where if the conditions of MVT are met AND f(a) = f(b), then f′(x) = 0 for some x with a ≤ x ≤ b
What is the extreme value theorem?
What is the difference between an absolute max/min and a local max/min?
If a function f(x) is continuous on a closed interval [a, b], then f(x) has both a maximum and minimum value on [a, b].
absolute max/min is the highest or lowest value within an interview, while local is still a max/min but does not have the highest or lowest value
Give a general strategy on how to solve related rates problem
draw a diagram
define variables
write what you know and what you want to find
relate variables together (usually through an equation)
find derivative and evaluate
What is the formula for linearization?
L(x) = f(a) + f'(a)(x-a)
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Determine all the number(s) c which satisfy the conclusion of Rolle’s Theorem for f(x)=x2−2x−8 on [−1,3].
Find critical points for f(x) = x3 -3x2 + 4 and on which intervals f(x) is increasing and decreasing.
x = 0, 2
(-infinity, 0) --> increasing; (0,2) --> decreasing; (2, infinity) --> increasing
Practice Exam 3 Q5
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f(x) = tan(x)cos(2x)
f'(x) =?
f'(x) = cos(2x)/cos2(x) + -2tan(x)sin(2x)
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Find the absolute maximum and the absolute minimum of the function f(x) = (x2−x+4)/x over the interval[1, 5].
absolute max f(5) = 4.8
absolute min f(2) = 3
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Find the derivative of (2)1/2 cos(x + y) = y2 and the line tangent to this curve at the point (pi/4 - 1, 1)
-(2)1/2sin(x+y) - y'((2)1/2sin(x+y)+2y)
y -1 = -(1/3)(x - pi/4+1)
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