Question
If $y=\frac{x-3}{2-5 x}$, then $\frac{d y}{d x}$ equals
(A) $\frac{17-10 x}{(2-5 x)^2}$
(B) $\frac{13}{(2-5 x)^2}$
(C) $\frac{-13}{(2-5 x)^2}$
(D) $\frac{17}{(2-5 x)^2}$
▶️Answer/Explanation
Ans:C
By the Quotient Rule (formula (6) on page 115),
$
\frac{d y}{d x}=\frac{(2-5 x)(1)-(x-3)(-5)}{(2-5 x)^2}
$
The Quotient Rule. $\lim _{x \rightarrow c}\left(\frac{f(x)}{g(x)}\right)=\frac{\lim _{x \rightarrow c} f(x)}{\lim _{x \rightarrow c} g(x)}=\frac{B}{D}$, provided that $D \neq 0$
Question
The maximum value of the function $f(x)=x e^{-x}$ is
(A) $\frac{1}{e}$
(B) 1
(C) $-1$
(D) $-e$
▶️Answer/Explanation
Ans:A
Here, $f(x)$ is $e^{-x}(1-x)$; $f$ has maximum value when $x=1$.
Question
$\int_0^2 \frac{x^2-3 x+7}{x+3} d x=$
(A) $\frac{4}{3}$
(B) $-10+25 \ln \frac{5}{3}$
(C) $2+7 \ln \frac{5}{3}$
(D) $\frac{32}{5} \ln 5$
▶️Answer/Explanation
Ans:B
(B) When we have a rational function in the integrand and the degree of the numerator is greater than the degree of the denominator, we need to rewrite the integrand using long polynomial division.
$
\begin{gathered}
\frac{-\left(x^2+3 x\right)}{-6 x+7} \\
\frac{-(-6 x-18)}{25} \\
\int_0^2\left(x-6+\frac{25}{x+3}\right) d x=\left.\left(\frac{x^2}{2}-6 x+25 \ln |x+3|\right)\right|_0 ^2 \\
=(2-12+25 \ln 5)-25 \ln 3=-10+25 \ln \frac{5}{3}
\end{gathered}
$
Question
Which equation has the slope field shown below?
(A) $\frac{d y}{d x}=\frac{5}{y}$
(B) $\frac{d y}{d x}=\frac{5}{x}$
(C) $\frac{d y}{d x}=\frac{x}{y}$
(D) $\frac{d y}{d x}=5 y$
▶️Answer/Explanation
Ans:A
Note that (1) on a horizontal line the slope segments are all parallel, so the slopes there are all the same and $\frac{d y}{d x}$ must depend only on $y ;(2)$ along the $x$-axis (where $y=0$ ) the slopes are infinite; and (3) as $y$ increases, the slope decreases.
Question
The graph below shows the velocity of an object moving along a line, for $0 \leq t \leq 9$.
At what time does the object attain its maximum acceleration?
(A) $2<t<5$
(B) $t=6$
(C) $t=8$
(D) $8<t<9$
▶️Answer/Explanation
Ans:D
Acceleration is the derivative (the slope) of velocity $v$; $v$ is largest on $8<t<9$.