Unit 1-2
lim(x ->0) [5x-10x2]/[ln(2x+1)+x]
5/3
What's the "order of a differential equation"?
the highest order of any derivative of the unknown function that appears in the equation.
If a1=4 and an=(an-1)2 + 3, then find the value of a3.
364
Whats the equation to find the nth degree Taylor polynomial?
Pn(x) = f(c)+f'(c)(x-c)+[(f''(c)/2!)(x-c)2+...[(f(n)(c)/n!)(x-c)n
A curve is defined by the parametric equations x(t) = -10t2+6t and y(t) = -4t3-8t2. Find dy/dx.
(-6t2-8t)/(-10t+3)
lim(x->2) [e4-2x-1]/[3ln(2x-3)]
-1/3
Given the differential equation dy/dx = x2/y2, find the particular solution, y = f(x) with the initial condition f(3)=-3.
y= cube root of (x3-54)
If a1=4 and an=2an-1 + 4, then find the value of a3.
28
What is the equation to find the nth degree Maclaurin Polynomial?
Pn(x) = f(c)+f'(c)(x)+[(f''(c)/2!)(x)2+...[(f(n)(c)/n!)(x)n
A curve is defined by the parametric equations x(t) = -4sin(-5t) and y(t) = -3e-8t. Find dy/dx.
(6e-8t)/(5cos(-5t))
Find the values of k that make the function continuous.
f(x)= k2+4x for x≥-2
f(x)= kx for x<-2
k = -4, 2
What is the particular solution to the differential equation dy/dx = -x/y with the initial condition y(-4) = 2?
y= sqrt of (-x2+20)
Evaluate the integral inf to 3 (-2/(x-2)2)dx or state the integral diverges (simplest form).
converges to -2
Write a degree 3 Taylor polynomial for f(x)=5sin(-2x) centered at x=-pi/2.
Write the equation of the line tangent to the graph of r=2sinθ when θ = pi/4.
x = 1
Find the value k that makes the function continuous.
f(x) = kx2 for x ≥2
f(x)= kx-8 for x<2
k = -4
Let y=f(x) be the solution to the differential equation dy/dx = -6xy-9 with initial condition f(4)=-6. What is the approximation for f(14) obtained by using the Euler's method with two step sizes of equal length, starting at x=4?
-180006
Evaluate the integral 2 to negative inf (ex)dx or state the integral diverges (simplest form).
converges to e2
Write a degree 2 Maclaurin Polynomial for f(x)=5cos(-2x).
A particle moving along a curve in the xy-plane has position (x(t),y(t)) = (5e10tsin(7t), 8e-3tcos(8t)), at time t≥0. Find the velocity vector of the particle at time t=0. Use a calculator.
(35, -24).
The function f(x) is continuous on its domain of [-9, 9] and is plotted below such that the portion of the graph on the interval (−3,2) is hidden from view. Given that f(−3)=5 and f(2)=7, determine what conclusions can be drawn based on the Intermediate Value Theorem on the interval (−3,2).
There may not exist a value of c in the interval (−3,2) where f(c)=3.
The rate of change dP/dt of the number of bears on an island is modeled by the following differential equation:
dP/dt = (1428/8833)P(1-P/924).
At t=0, the number of bears on the island is 121 and is increasing at a rate of 17 bears per day. Assuming t≥0, for what values of P is d2P/dt2 >0?
P(t) is concave up when 121<P<462.
Evaluate the integral 2 to negative inf (-4e2x)dx or state the integral diverges (simplest form).
converges to -2e4
Write a degree 5 Maclaurin Polynomial for f(x)=4sin(-4x).
The velocity of a particle moving in the xy-plane is given by the vector v(t) = (-3t3, -4ln(t+8)). At time t=0, the position of the particle is <10,0>. Find the position of the particle at time t=2. Round to 3 decimal places if necessary.
<-2, -17.561>