Question

For the following functions,

a. use Equation 3.4 to find the slope of the tangent line m_tan = f' (a), and
b. find the equation of the tangent line to f at x = a.
f(x)=3-4x, a = 2
f(x) =x/5 +6, a = -1

Answer 1

To find the slope of the tangent line, take the derivative of the function and evaluate it at the given point. The equation of the tangent line can be found using the point-slope form.

Step-by-step explanation:

To find the slope of the tangent line using Equation 3.4, we need to take the derivative of the function at the given point.

a. For the function f(x) = 3 - 4x and a = 2, we find the derivative f'(x) = -4. Therefore, the slope of the tangent line at x = 2 is m_tan = f'(2) = -4.

b. To find the equation of the tangent line, we use the point-slope form y - y1 = m(x - x1), where (x1, y1) is the point of tangency. Plugging in the values x1 = 2, y1 = f(2) = 3 - 4(2) = -5, and m = -4, we get the equation y + 5 = -4(x - 2).

For the function f(x) = x/5 + 6 and a = -1, the derivative f'(x) = 1/5. Therefore, the slope of the tangent line at x = -1 is m_tan = f'(-1) = 1/5. Using the same point-slope form, we plug in x1 = -1, y1 = f(-1) = -1/5 + 6 = 29/5, and m = 1/5 to get the equation y - 29/5 = 1/5(x + 1).