Derivatives
Other Derivatives
Applications of the Derivative
Related Rates
Miscellaneous
300
Differentiate:
log_5(2^(cospix))
-pilog_(5)2sin(pix)
300
Find y'. Make sure your final answer is simplified appropriately and only in terms of x and y.
ln(xy) - 2y^2 = 5x - 2
y' = (5xy-y)/(x - 4xy^2)
300
Consider the picture below. A particle is moving along a horizontal axis (in meters) over time (in seconds). At what time(s) is the particle changing direction if...
a) The graph represents the position function
b) The graph represents the velocity function
a) t = 1, 3, 5 seconds
b) t = 1.5, 4 seconds
300
The angle of elevation is the angle formed by a horizontal line and a line joining the observer’s eye to an object above the horizontal line. A person is 500 feet way from the launch point of a hot air balloon. The hot air balloon is starting to come back down at a rate of 15 ft/sec. At what rate is the angle of elevation, θ, changing when the hot air balloon is 200 feet above the ground. Your final answer should be exact and simplified (no decimals!)
-3/116
radians/sec
300
lim_(x->0)((4x^2)/sin^2(5x))
4/25
400
Find y'
y = 3/(sin^3(cos(3x))
y' = (-27sin(3x)cos(cos(3x)))/sin^4(cos(3x))
400
Find the first derivation. Make sure your answer is explicit.
f(x) = (lnx)^(sinx)
y' = (lnx)^sinx((sinx)/(xlnx) + ln(lnx)cosx)
400
The position of a particle moving along a vertical axis, in feet, over t seconds is given by
s(t) = 1.5t^3-13.5t^2 +22.5t
Find the total distance traveled over the first 6 seconds
69 feet
400
Bike A is 45 miles west of an intersection and moving west at 20 miles/hour. Bike B is 15 miles south of the same intersection and moving north at 15 miles/hour. At what rate is the distance between the bikes changing at that point in time? Please provide an exact answer and is simplified and rationalized. Don't forget units!
(9sqrt10)/2 mph
400
Consider the function h below. f and h are inverses, and f(-2) = -14.
f(x) = 1/2x^3 + 3x - 4
Find
h'(-14)
1/9
500
Find f'(x)
f(x) = tan^-1sqrt(5-sec^2(3 - x))
(sec^2(3 - x)tan(3 - x))/((6 - sec^2(3 - x))sqrt(5 - sec^2(3 - x))
500
Find y'. Please make sure your answer is strictly in terms of x and y and is simplified, if possible.
y = (5xy)/(3 - 4xy^2)
y' = (15y)/(9 - 24xy^2 + 16x^2y^4-20x^2y^2- 15x)
500
The position function of a particle moving horizontally over x seconds is given by:
s(t) = 2sin(x/2)
Describe where the particle is speeding up and slowing down over the first 15 seconds.
Speeding Up:
(pi, 2pi), (4pi, 5pi) seconds
Slowing Down:
(0, pi), (2pi, 3pi), (4pi, 15) seconds
500
A tank of water in the shape of a cone is being filled with water at a rate of 12 m3/sec. The circumference of the top of the tank is 52(pi) meters and the height of the tank is 8 meters. At what rate is the depth of the water in the tank changing when the radius of the top of the water is 10 meters?
3/(25pi) m/sec
500
Consider the function:
f(x) = (x + 2)^3
Create the Linearization Function, L(x), that you would use to approximate (4.1)3. Then, use L(x) to provide a simplified rational approximation for (4.1)3. Include L(x) and the approximation as part of your final answer.
L(x) = 48x - 32
344/5