91Ó°ÊÓ

Find the general antiderivative \(F(x)+C\) for each of the following. $$ f(x)=x-4 $$

, 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. $$ g(x)=|x| ;[-2,2] $$

Solve the given differential equation subject to the given condition. Note that \(y(a)\) denotes the value of \(y\) at \(t=a\). $$ \frac{d y}{d t}=6 y, y(0)=1 $$

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. $$ x^{4}+5 x^{3}+1=0 ;[-1,0] $$

Use the Monotonicity Theorem to find where the given function is increasing and where it is decreasing. $$ g(x)=(x+1)(x-2) $$

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. $$ -x \frac{d y}{d x}+y=0 ; y=C x $$

Find the general antiderivative \(F(x)+C\) for each of the following. $$ f(x)=x^{2}+\pi $$

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. $$ f(\theta)=\sin 2 \theta, 0<\theta<\frac{\pi}{4} $$

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. $$ \frac{d^{2} y}{d x^{2}}+y=0 ; y=C_{1} \sin x+C_{2} \cos x $$

Use the Monotonicity Theorem to find where the given function is increasing and where it is decreasing. $$ h(t)=t^{2}+2 t-3 $$