Limits
Evaluate lim_{x --> 0} |x|.
0
Is f(x) = |x| continuous everywhere?
Yes
Differentiate (x^3 + x^2 + sqrt(x))/x.
2x + 1 - (1/2)x^(-3/2)
Is dy/dx = x autonomous?
No
Integrate (x + 2)(x - 1) from 0 to 1
-7/6
Evaluate lim_{x --> 0} sin(x)/x.
1
Does ln[ sin(x)^7 ] have a maximum and minimum on the interval [pi/3, pi/2]? Why?
Yes. By EVT.
Differentiate 1/[log(log(x))]
-1/[log(log(x))]^2 * 1/log(x) * 1/x
What are all the equilibrium solutions of dy/dx = xy?
y(x) = 0
What is the indefinite integral of (e^x)^23?
(1/23)*e^(23x)
Evaluate lim_{x --> 1} 1/(x - 1)^2.
+infinity
Consider the function f(x) = e^(-1/x^2) for x not 0, and 0 for x = 0.
Is the function continuous?
Yes because lim_{x --> 0} f(x) = 0 = f(0) so it is continuous at x = 0. It is also continuous at other values because there the function is a composition of continuous functions.
If the area of a circle is increasing at a rate of 10pi,
then how fast is the circumference increasing when the radius is r = 5.
dC/dt = 2pi when r = 5.
Is the equilibrium solution y(x) = 1 for dy/dx = y(1 - y) stable or unstable.
Stable
What is the antiderivative of ln(x)?
x(ln(x) - 1)
Evaluate lim_{x --> 0} (e^x - 1)/ln(1 + x).
1 by L'Ho^pital's rule.
Explain why every odd degree polynomial has a root (i.e. there is a c such that p(c) = 0).
Because lim_{x --> infinity} p(x) = infinity
and lim_{x --> -infinity} p(x) = -infinity
so it follows by IVT.
Maximize profit if the price function is p(x) = 20 - x and the cost function is C(x) = 50. What is the maximum profit and what should the price of the product be to achieve this?
Maximum profit is 50 which is attained when the price is p(10) = 10.
Consider the ODE dy/dx = y(y - 1)(y - 2)(y - 3).
What is lim_{x --> infinity} y(x) for the solution y(x) with initial conditions y(0) = 1.5?
2
Evaluate
lim_{x --> 0} (integral of sin(cos(t)) from t = 0 to x)/x.
sin(1)