PURE Differentiation
sin(ln(x))
(cos(ln(x)))/x
The altitude of a triangle is increasing at a rate of 1 cm/min while the area of the triangle is increasing at a rate of 2 cm2/min. At what rate is the base of the triangle changing when the altitude is 10 cm and the area is 100 cm2?
-1.6 cm/min
What is the velocity after 1 second?
-1/2*e-3/2
Find the intervals on which f is increasing.
(-1,1)
A Fart Heard Around the World was the headliner of this former presidential candidate last week.
Eric Swalwell
ln(xe-2x)
(-2e-2xx+e-2x)/(xe-2x)
A spotlight on the ground shines on a wall 12 m away. If a man 2 m tall walks from the spotlight toward the building at a speed of 1.6 m/s, how fast is the length of his shadow on the building decreasing when he is 4 m from the building?
-0.6 m/s
When is the particle at rest?
2 seconds
Find the local maximum value(s) of f.
y=3
The NFL disciplined 33 players from the Pittsburgh Steelers and Cleveland Browns over a fight that broke out. Which Browns player hit another man in the head with his helmet?
Myles Garrett
(x2+1)3(x2+2)6
6x(x2+1)2(x2+2)6+12x(x2+1)3(x2+2)5
A kite 100 ft above the ground moves horizontally at a speed of 8 ft/s. At what rate is the angle between the string and the horizontal decreasing when 200 ft of string has been let out?
0.02 radians/second or 1.15 degrees/second
When is the particle moving in the positive direction?
t<2 or [0,2)
Find the inflection points.
-sqrt(3),0,sqrt(3)
At the AMA's on Sunday, Taylor Swift took home 6 ama awards, making this her new ama award count
29
(t2)/(sqrt(t3+1))
See key
A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m higher than the bow of the boat. If the rope is pulled in at a rate of 1 m/s, how fast is the boat approaching the dock when it is 8m from the dock?
sqrt(65)/8 m/s
Find the total distance traveled within the first 6 seconds.
1.325 ft
Find the intervals of when it is concave down.
(-infinity,-sqrt(3)) U (0,sqrt(3))
What are the conditions that need to be met for Mean Value Theorem?
1. Continuous in the closed interval [a,b]
2. Differentiable in the open interval (a,b)
(tan(5x3)+1)/(e-x)
See board
Pg. 250 #25
289,000 cm3/min
When is the particle speeding up?
(2,4)
Pg. 301 #28
Increasing everywhere except at the vertical asymptote at x=1. Concave up below x=1 and above x=3 and concave down between x=1 and x=3
What is the conclusion for Mean Value Theorem?
There exists a c between a and b such that f'(c)=f(b)-f(a)/b-a