Problem 1
In Problems \(1-4\), show that the indicated function is a solution of the given differential equation; that is, substitute the indicated function for \(y\) to see that it produces an equality. $$ \frac{d y}{d x}+\frac{x}{y}=0 ; y=\sqrt{1-x^{2}} $$
Problem 1
Use the Bisection Method to approximate the real root of the given equation on the given interval. Each answer should be accurate to two decimal places. $$ x^{3}+2 x-6=0 ;[1,2] $$
Problem 1
Find the general antiderivative \(F(x)+C\) for each of the following. $$ f(x)=5 $$
Problem 1
Identify the critical points. Then use (a) the First Derivative Test and (if possible) (b) the Second Derivative Test to decide which of the critical points give a local maximum and which give a local minimum. $$ f(x)=x^{3}-6 x^{2}+4 $$
Problem 1
In Problems \(1-4\), solve the given differential equation subject to the given condition. Note that \(y(a)\) denotes the value of \(y\) at \(t=a\). $$ \frac{d y}{d t}=-6 y, y(0)=4 $$
Problem 1
, a function is defined and a closed interval is given. Decide whether the Mean Value Theorem applies to the given function on the given interval. If it does, find all possible values of \(c ;\) if not, state the reason. In each problem, sketch the graph of the given function on the given interval. $$ f(x)=|x| ;[1,2] $$
Problem 1
In Problems \(1-10\), use the Monotonicity Theorem to find where the given function is increasing and where it is decreasing. $$ f(x)=3 x+3 $$
Problem 1
Find two numbers whose product is \(-16\) and the sum of whose squares is a minimum.
Problem 2
Find the general antiderivative \(F(x)+C\) for each of the following. $$ f(x)=x-4 $$
Problem 2
Use the Monotonicity Theorem to find where the given function is increasing and where it is decreasing. $$ g(x)=(x+1)(x-2) $$
