Question
If \(f(x)=\frac{e^2x}{2x}\), then \(f'(x)\)=
(A) 1
(B)\(\frac{e^2x(1-2x)}{2x^2}\)
(C)\(e^2x\)
(D)\(\frac{e^2x(2x+1)}{x^2}\)
(E)\(\frac{e^2x(2x-1)}{2x^2}\)
▶️Answer/Explanation
Ans:E
Question
The temperature of a solid at time t ≥ 0 is modeled by the nonconstant function H and increases according to the differential equation \(\frac{dH}{dt}=2H+1\) , where H(t) is measured in degrees Fahrenheit and t is measured in hours. Which of the following must be true?
A \(H=H^2+ t + C\)
B \(ln |2H+1| = t/2 + C\)
C \(ln |2H+1| = t + C\)
D \(ln |2H+1| = 2t + C\)
▶️Answer/Explanation
Ans:D
Question
At each point (x , y ) on a certain curve, the slope of the curve is \(3x^{2} y \). If the curve contains the point ( 0,8) , then its equation is
(A)\(y=8e^{x^{3}}\) (B)\(y=x^{3}+8\) (C)\(y=e^{x^{3}}+7\) (D)\(y=In(x+1)+8\) (E)\(y^{2}=x^{3}+8\)
▶️Answer/Explanation
Ans:A
Question
The general solution of the differential equation \(y’=y+x^{2} \)is y=
(A)\(Ce^{x}\) (B)\(Ce^{x}+x^{2}\) (C)\(-x^{2}-2x-2+C (D)e^{x}-x^{2}-2x-2+C\) (E)\(Ce^{x}-x^{2}-2x-2\)
▶️Answer/Explanation
Ans:E
No longer covered in the AP Course Description. The solution is of the form \(y=Y_hY_p\) where \(Y_h\) is the solution to y’- y = 0 and the form of\( y_{p},\) is Ax2 + Bx+K . Hence\( y = Ce^{x}\). Substitute \(У_р\) into the original differential equation to determine the values of A, B, and K.Another technique is to substitute each of the options into the differential equation and pick the one that works. Only (A), (B), and (E) are viable options because of the form for \(y_h\) Both (A) and (B) fail, so the solution is (E).