Derivatives
f(x) = x2⋅sin(x)
Find f'(x).
2xsin(x) + x2cos(x)
Find dy/dx.
y2 = x3+y
3x2/(2y-1)
A square metal plate is heated, causing its side length to expand at a rate of 0.02 cm/min. At the moment when the side length is 10 cm, how fast is the area of the plate changing?
The area of the plate is increasing at 0.4 cm²/min.
Find the linear approximation FORMULA [L(x)] of f(x)=sqrt(4) at x=4
L(x) = 2 + 1/4(x-4)
A population of bacteria is modeled by the function P(t)=200e0.05t, where t is in hours. Give the population equation after 10 hours
P(10) = 200e0.5
f(x) = (3x2+1)/x
Find f'(x).
(3x2-1)/x2
Find dy/dx.
sin(y)=x2+2y
2x/cos(y)-2
A circle's radius is increasing at a rate of 2 cm/s. Find the rate at which the circle's area is changing when the radius is 5 cm.
The area is increasing at 20π cm²/s.
Use linearization to approximate sqrt(49.5)
= 7.036 (approx)
A radioactive substance decays according to the function A(t)=100e−0.03t, where t is in years. Find the amount remaining after 25 years
A(25) = 47.24 (approx)
f(x) = (2x3+5)4
Find f'(x).
24x2(2x3+5)3
Find dy/dx.
x2+y2=25
-x/y
Air is being pumped into a spherical balloon at a rate of 100 cm³/s. Find the rate at which the radius is changing when the radius is 10 cm.
The radius is increasing at 1/4π cm/s.
Find the linear approximation of f(x)=ex/2 at x=0, then use it to approximate e0.1
= 1.05 (approx)
The half-life of a certain substance is 5 years. If the initial amount is 200 grams, how much remains after 12 years?
38 grams (approx)
f(x) = tan(x2)
Find f'(x).
2x⋅sec2(x2)
Find dy/dx.
ey=x2+y3
2x/(ey-3y2)
A ladder 10 ft long is leaning against a wall. If the bottom of the ladder slides away from the wall at 2 ft/s, how fast is the top of the ladder sliding up or down the wall when the bottom is 6 ft from the wall?
The top of the ladder is sliding down at 3/2 ft/s
Approximate cos(0.1) using the linearization of f(x)=cos(x) at x=0
= 1 (approx)
A population doubles every 6 years. If the initial population is 500, how large will the population be after 20 years?
The population after 20 years is approximately 5040
f(x) = ln(x2+1)⋅e3x
Find f'(x).
2x/(x2+1)⋅e3x + ln(x2+1)⋅3e3x
Find dy/dx.
cos(xy)=x+y
[y⋅sin(xy)+1]/-(x⋅sin(xy)+1)
A conical tank is being filled with water at a rate of 5 m³/min. The tank has a height of 12 m and a radius of 4 m. Find the rate at which the water level rises when the water is 3 m deep.
The water level is rising at 5/π m/min.
The temperature T (in °C) of a metal rod t minutes after being placed in a room follows the equation T(t)=150e-0.2t+20. Use linearization at t=0 to approximate the temperature after 1 minute
T(1) = 140 C (approx)
A certain bacteria population follows the exponential growth model P(t)=800e0.04t, where P(t) is the population at time t in hours. After how many hours will the population reach 150% of its initial size?
10.14 years (approx)