Derivatives/Integrals
If y = e^−x^2, then y"(0) =
-2
The population of trout in a stocked pool is given by F(t) = 60 − 20 cos (π/12t), where t is
measured in months since January. Find the average number of trout in the pond from May
through August.
57 Trout
y = tan^−1(cosx)
y′ =−sinx/1+cos^2
Solve dy/dx= x + 1 for a general solution
y=1/2x^2+x+c
∫3e^x dx
3e^x+c
The temperature (in °F) t hours after 9 AM is approximated by the function
T(t) = 50 + 14 sin (πt/12). Find the average temperature during the time period 9 AM to 9PM
58.913°F
f(x) = x ln(tan−1 x)
ln(tan−1 x) +x/(1+x2)tan^−1 x
Consider the differential equation dy/dx= (y − 2)(x^2 + 1). Find y = g(x), the particular solution to
the given differential equation with initial condition g(0) = 5.
y = 3e^1/3x^3+x + 2
y = −3 ln(x^2)
y'=-6/x
Suppose that the velocity function of a particle moving along a coordinate line is
v(t) = 3t^3 + 2. Find the average velocity of the particle over the time interval 1 ≤ t ≤ 4.
263/4
∫12/1+9x^2 dx
4 arctan(3x) + C
Find the solution to the differential equation dy/dx=sinx/e^y, where y (π/4) = 0.
y = ln (−cos x +√2/2+ 1)
y = sin(ln x)
y'=cos(lnx)/x
What is the average value of y =cos x /x^2+x+2
on the closed interval [-1, 3]?
0.183
y = xsin^−1(x)
y′ = sin^−1(x) +x/√1−x^2
If dy/dx=1/4y and y(0) = 5, then y(4) =
5e
∫x/9-x^2 dx
−1/2ln|9 − x^2| + C
The average value of the function f(x) = ln2x on the interval [2, 4] is.....
1.204
If f(x) = sin−1 x, then f′ (√3/2) =
2
Consider the differential equation dy/dx= (1 − 2x) ∙ y. If y = 10 when x = 1, find an equation for y.
y = 10e^x−x^2