Question

The line tangent to the graph of g(x) = x^{3}-4x+1 at the point (2, 1) is given by the formula

L(x) = 8(x - 2) + 1.


Which of the following statements are true? Select all that apply.
A. The functions are approximately equal on the interval 1.9 smaller than or equal to x larger than or equal to 2.1.
B. The functions have the same y-intercept.
C. The slope of the line equals g'(0).
D. The functions are approximately equal on the interval 0.5 smaller than or equal to x larger than or equal 1.0.
E. The graphs intersect at the point (2,1).
F. The slope of the line equals g'(2).
G. None of the above are true.

Answer 1

well, about A and D, I just plugged the values on the slope formula of

$\bf \begin{array}{llll} g(x)=x^3-4x+1\\ L(x) = 8(x-2)+1 \end{array} \qquad \begin{cases} x_1=1.9\\ x_2=2.1 \end{cases}\implies \cfrac{f(b)-f(a)}{b-a}$

for A the values are 8.01 and 8.0, so indeed those "slopes" are close. $\textit{\huge \checkmark}$

for D the values are -2.25 and 8.0, so no dice on that one.

for B, let's check the y-intercept for g(x), by setting x = 0, we end up g(0) = 0³-4(0)+1, which gives us g(0) = 1.

checking L(x) y-intercept, well, L(x) is in slope-intercept form, thus the +1 sticking out on the far right is the y-intercept, so, dice. $\textit{\huge \checkmark}$

for C, well, the slope if L(x) is 8, since it's in slope-intercept form, the derivative of g(x) is g'(x) = 3x² - 4, and thus g'(0) = -4, so no dice.

for E, do they intercept at (2,1)?  well, come on now, L(x) is a tangent line to g(x), so that's a must for a tangent. $\textit{\huge \checkmark}$

for F, we know the slope of the line L(x) is 8, is g'(2) = 8?  let's check

recall that g'(x) = 3x² - 4, so g'(2) = 3(2)² - 4, meaning g'(2) = 8, so, dice. $\textit{\huge \checkmark}$

Answer 2

Using function concepts, it is found that the correct options are:

• A. The functions are approximately equal on the interval 1.9 smaller than or equal to x larger than or equal to 2.1.
• E. The graphs intersect at the point (2,1).
• F. The slope of the line equals g'(2).

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• The value of x we are interested is x = 2, thus, considering a margin, the functions are approximately equal between x = 1.9 and x = 2.1, which means that option A is correct, while option D is not.
• The y-intercept of g is $g(0) = 0^3 - 4(0) + 1 = 1$, while the y-intercept of L is $L(0) = 8(0 - 2) + 1 = -16 + 1 = - 15$. Not the same, thus, option B is not correct.
• The slope of the tangent line to g at x = 2 is the derivative of g at x = 2, that is, g'(2), thus option F is correct while option C is not.
• At (2,1), the g(x) and L(x) intersect, thus, option E is correct.

A similar problem is given at https://brainly.com/question/22426360