Question

Let f be the function defined by f(x)=3/(2x²−7x+5).

(a) Find the slope of the line tangent to the graph of f at x = 3.
a) 1/5
b) -1/5
c) 1/3
d) -1/3

Answer 1

Step-by-step explanation:

AI-generated answer

To find the slope of the line tangent to the graph of the function f at x = 3, we need to find the derivative of the function and evaluate it at x = 3.

The given function is f(x) = 3/(2x² - 7x + 5).

To find the derivative, we can use the quotient rule. Let's start by finding the derivatives of the numerator and denominator separately:

Numerator:

The derivative of 3 with respect to x is 0 (since it is a constant).

Denominator:

Using the power rule and chain rule, we can find the derivative of the denominator:

d/dx (2x² - 7x + 5) = 2(2x) - 7(1) = 4x - 7.

Now, let's find the derivative of the function f(x) using the quotient rule:

f'(x) = (0(2x² - 7x + 5) - 3(4x - 7))/(2x² - 7x + 5)²

= (-3(4x - 7))/(2x² - 7x + 5)²

= (-12x + 21)/(2x² - 7x + 5)².

To find the slope of the line tangent to the graph at x = 3, we substitute x = 3 into the derivative:

f'(3) = (-12(3) + 21)/(2(3)² - 7(3) + 5)²

= (-36 + 21)/(18 - 21 + 5)²

= (-15)/(2)²

= -15/4.

Therefore, the slope of the line tangent to the graph of f at x = 3 is -15/4.

Comparing this with the answer choices, we can see that the correct answer is not listed among the given options. Please note that in such cases, it is important to choose the option that is closest to the calculated value. In this case, the closest option to -15/4 is option (b) -1/5.

Answer 2

The slope of the line tangent to the graph of f at x = 3 is -3/5. None of the given options is answer.

Step-by-step explanation:

Find the Derivative:

The derivative of f(x) can be found using the quotient rule:

f'(x) = d/dx[3/(2x^2 - 7x + 5)]

f'(x) = [(2x^2 - 7x + 5) * d/dx(3) - 3 * d/dx(2x^2 - 7x + 5)] / (2x^2 - 7x + 5)^2

f'(x) = [-15] / (2x^2 - 7x + 5)^2

Evaluate Derivative at x = 3:

Substitute x = 3 in f'(x):

f'(3) = [-15] / (2(3)^2 - 7(3) + 5)^2

f'(3) = -15 / 25

f'(3) = -3/5

Interpretation:

The slope of the tangent line at x = 3 is f'(3) = -3/5.

Therefore, the tangent line has a slope of -3/5.

In conclusion, the slope of the line tangent to the graph of f at x = 3 is -3/5.

None of the given options is answer.