Problem 36
You should solve the following problems without using a graphing calculator. True or False If \(f^{\prime}(x)=g(x),\) then \(\int x g(x) d x=\) \(x f(x)-\int f(x) d x .\) Justify your answer.
Problem 36
Limited Growth Equation Another differential equation that models limited growth of a population \(P\) in an environment with carrying capacity \(M\) is \(d P / d t=k(M-P)\) (where \(k>0\) and \(M>0\) ). (a) Show that \(P=M-A e^{-k t},\) where \(A\) is a constant determined by an appropriate initial condition. (b) What is lim \(P(t) ? ~ M\) (c) For what time \(t \geqslant 0\) is the population growing the fastest? (d) Writing to Learn How does the growth curve in this model differ from the growth curve in the logistic model? See answ
Problem 37
In Exercises \(29-34\) , construct a slope field for the differential equation. In each case, copy the graph at the right and draw tiny segments through the twelve lattice points shown in the graph. Use slope analysis, not your graphing calculator. $$\frac{d y}{d x}=x-y$$
Problem 37
In Exercises \(25-46,\) use substitution to evaluate the integral. $$\int \frac{\sin (2 t+1)}{\cos ^{2}(2 t+1)} d t$$
Problem 37
True or False If \(f^{\prime}(x)=g(x),\) then \(\int x^{2} g(x) d x=\) \(x^{2} f(x)-2 \int x f(x) d x .\) Justify your answer.
Problem 38
A savings account earning compound interest triples in value in 10 years. How long will it take for the original investment to quadruple?
Problem 38
Multiple Choice If \(\int x^{2} \cos x d x=h(x)-\int 2 x \sin x d x,\) then \(h(x)=\) (A) \(2 \sin x+2 x \cos x+C\) (B) \(x^{2} \sin x+C\) (C) \(2 x \cos x-x^{2} \sin x+C\) (D) \(4 \cos x-2 x \sin x+C\) (E) \(\left(2-x^{2}\right) \cos x-4 \sin x+C\)
Problem 38
In Exercises \(25-46,\) use substitution to evaluate the integral. $$\int \frac{6 \cos t}{(2+\sin t)^{2}} d t$$
Problem 39
You should solve the following problems without using a graphing calculator. True or False For small values of \(t\) the solution to logistic differential equation \(d P / d t=k P(100-P)\) that passes through the point \((0,10)\) resembles the solution to the differential equa- tion \(d P / d t=k P\) that passes through the point \((0,10) .\) Justify your answer.
Problem 39
The Inversion of Sugar The processing of raw sugar has an "inversion" step that changes the sugar's molecular structure. Once the process has begun, the rate of change of the amount of raw sugar is proportional to the amount of raw sugar remaining. If 1000 \(\mathrm{kg}\) of raw sugar reduces to 800 \(\mathrm{kg}\) of raw sugar during the first \(10 \mathrm{h},\) how much raw sugar will remain after another 14 \(\mathrm{h}\) ?
