Question

1. the top of a 25 - foot ladder is sliding down a vertical wall at a constant rate of 3 feet per minute. when the top of the ladder is 7 feet from the ground, what is the rate of change of the distance between the bottom of the ladder and the wall? (a) -\\(\frac{7}{8}\\) feet per minute (b) -\\(\frac{9}{14}\\) feet per minute (c) \\(\frac{9}{14}\\) feet per minute (d) \\(\frac{7}{8}\\) feet per minute (e) \\(\frac{21}{16}\\) feet per minute 6. the line tangent to the graph of the twice - differentiable function (f) at the point (x = 3) is used to approximate the value of (f(3.25)). which of the following statements guarantees that the tangent line approximation at (x = 3.25) is an underestimate of (f(3.25))? (a) the function (f) is decreasing on the interval (3\leq x\leq3.25). (b) the function (f) is increasing on the interval (3\leq x\leq3.25). (c) the graph of the function (f) is concave down on the interval (3\leq x\leq3.25). (d) the graph of the function (f) is concave up on the interval (3\leq x\leq3.25). 7. the locally linear approximation of the differentiable function (f) at (x = 3) is used to approximate the value of (f(3.2)). the approximation at (x = 3.2) is an overestimate of the corresponding function value at (x = 3.2). which of the following could be the graph of (f)?

Explanation:

Step1: Analyze ladder - related problem

Let the height of the top of the ladder on the wall be $y$ and the distance between the bottom of the ladder and the wall be $x$. By the Pythagorean theorem, $x^{2}+y^{2}=25^{2}=625$. Differentiating both sides with respect to time $t$, we get $2x\frac{dx}{dt}+2y\frac{dy}{dt}=0$, which simplifies to $x\frac{dx}{dt}+y\frac{dy}{dt}=0$. We know that $\frac{dy}{dt}=- 3$ (negative because $y$ is decreasing). When $y = 7$, we can find $x$ using $x=\sqrt{625 - y^{2}}=\sqrt{625 - 49}=\sqrt{576}=24$. Substituting $x = 24$, $y = 7$ and $\frac{dy}{dt}=-3$ into $x\frac{dx}{dt}+y\frac{dy}{dt}=0$, we have $24\frac{dx}{dt}+7\times(-3)=0$.

Step2: Solve for $\frac{dx}{dt}$

From $24\frac{dx}{dt}-21 = 0$, we can solve for $\frac{dx}{dt}$:
\[

$$\begin{align*} 24\frac{dx}{dt}&=21\\ \frac{dx}{dt}&=\frac{21}{24}=\frac{7}{8} \end{align*}$$

\]

Step3: Analyze tangent - line approximation problem

The tangent - line approximation of a function $y = f(x)$ at $x=a$ is given by $L(x)=f(a)+f^{\prime}(a)(x - a)$. If the function $f$ is concave up on the interval $[a,x]$, then the tangent - line approximation $L(x)$ is an underestimate of $f(x)$. If the function $f$ is concave down on the interval $[a,x]$, then the tangent - line approximation $L(x)$ is an overestimate of $f(x)$.

Answer:

1. D. $\frac{7}{8}$ feet per minute
2. D. The graph of the function $f$ is concave up on the interval $3\leq x\leq3.25$