Find an equation of the tangent line to the graph of the following function f at the specified point.

Question

f(x) = (x³ + 5)(3x² − 4x + 2); (1, 6)

Answer 1

The equation of a tangent line is y = f'(x₁) (x - x₁) + f(x₁)

--> so we need to find f'(x₁), which is the derivative of the original function at the x-value provided (x₁ = 1),

and then plug in the given point (1, 6) for the x₁ and f(x₁) values.

First, let's simplify the f(x) function by distributing:

f(x) = (x³ + 5)(3x² - 4x + 2)

Distribute the x³ to all three terms in the second group AND distribute the 5 to all three terms in the second group:

= x³(3x² - 4x + 2) + 5(3x² - 4x + 2)

When you multiply terms with exponents, add the exponents together:

= 3x⁵ - 4x⁴ + 2x³ + 15x² - 20x + 10

Now, let's find the general derivative of f(x) using the Power Rule (multiply the front number of each term by the power and then reduce the power by 1):

f'(x) = 5*3x⁴ - 4*4x³ + 3*2x² + 2*15x¹ - 1*20x⁰ + 0

(the derivative of a number without a variable is just 0)

Now simplify:

f'(x) = 15x⁴ - 16x³ + 6x² + 30x - 20 = the general derivative

(x⁰ becomes the number 1)

Now let's plug in x = 1 to find f'(1):

f'(1) = 15*1⁴ - 16*1³ + 6*1² + 30*1 - 20

f'(1) = 15 - 16 + 6 + 30 - 20

f'(1) = 15

This is what you will plug in for f'(x₁).

So, we have all the pieces we need:

y stays "y"

f(x₁) = 6 (the given y-value)

f'(x₁) = 15

x stays "x"

x₁ = 1 (the given x-value)

Let's plug them into the tangent line equation:

y = f'(x₁) (x - x₁) + f(x₁)

y = 15 (x - 1) + 6

Then distribute and simplify:

y = 15x - 15 + 6

y = 15x - 9

The end!