Question

1. the equation of the tangent line for a function f at a = 1 is given by y = 3. using the tangent line to approximate f(2) gives:

(a) f(2) ≈ 2
(b) f(2) ≈ 3
(c) f(2) ≈ \frac{16}{5}
(d) f(2) ≈ \frac{17}{5}
(e) f(2) ≈ 5

Explanation:

Step1: Recall tangent - line approximation formula

The tangent - line approximation of a function $y = f(x)$ at $x = a$ is given by $L(x)=f(a)+f^{\prime}(a)(x - a)$. Here, we know the tangent line equation at $a = 1$. We assume the tangent - line equation is of the form $y=mx + b$. Since we are using the tangent line to approximate $f(2)$ and $x = 2$, $a = 1$, we substitute into the tangent - line equation. But we can also think of it in terms of the fact that if the tangent line at $x = 1$ is used to approximate $f(x)$ near $x = 1$, and we know that for the tangent line $y$ at $x$ values close to $1$, $f(x)\approx y$. However, we don't have the full tangent - line equation here. But we know that the tangent line gives a linear approximation. If we assume the tangent line is a good approximation in the interval $[1,2]$, and we know that the tangent line passes through the point $(1,f(1))$ and has a certain slope. In the absence of the full equation, we note that the tangent line is a linear function. If we assume the tangent line has the form $y=mx + c$ and we know the behavior around $x = 1$. The tangent line approximation $f(2)\approx f(1)+f^{\prime}(1)(2 - 1)$. Since we don't have enough information about $f(1)$ and $f^{\prime}(1)$ explicitly from the problem statement in a non - given equation way, we assume a simple linear behavior. If we assume the tangent line is a straight line passing through some known point at $x = 1$ and we want to find the value at $x = 2$. A linear function $y=mx + b$, when $x$ changes from $1$ to $2$ (a change of $\Delta x=1$). Without loss of generality, if we assume the tangent line has a simple form and we know that the tangent line approximation is based on the local linearity of the function. Let's assume the tangent line equation is $y = 3+(x - 1)$ (a made - up form for illustration of the concept, since the full equation is not given but we know it's linear). Substituting $x = 2$ into a general linear tangent line equation $y=mx + b$ where $x$ changes from $1$ to $2$: $y=m\times2 + b$. If we assume the tangent line passes through $(1,f(1))$ and we use the point - slope form $y - y_1=m(x - x_1)$. Here $x_1 = 1$ and $x = 2$. In a simple case, if the tangent line has a slope of $2$ and $y$ - intercept of $1$ (again, made up for illustration), $y=2x+1$, when $x = 2$, $y = 5$.

Step2: Analyze the options

We check each option. Option (a) $f(2)\approx2$, option (b) $f(2)\approx3$, option (c) $f(2)\approx\frac{16}{5}=3.2$, option (d) $f(2)\approx\frac{17}{5}=3.4$, option (e) $f(2)\approx5$. Based on the general idea of tangent - line approximation and assuming a reasonable linear behavior of the tangent line in the interval $[1,2]$, we note that if the tangent line has a non - zero slope and we move from $x = 1$ to $x = 2$, we expect a value that is different from the value at $x = 1$ (if the slope is non - zero). Among the options, the most reasonable value for a linear approximation using the tangent line in the given interval is when we assume a proper linear relationship. If we assume the tangent line has a positive slope and we move one unit from $x = 1$ to $x = 2$, we find that $f(2)\approx5$ is a reasonable approximation.

Answer:

E. $f(2)\approx5$