Question

Let f(x)=7 x²-3 x.

(a) Find the derivative of the line tangent to the graph of f at x=4.
Slope at x=4 :
(b) Find an equation of the line tangent to the graph of f at z=4.
Tangent line: y=

Answer 1

The derivative of f(x) = 7x^2 - 3x is 14x - 3. The slope of the tangent line at x = 4 is 53. The equation of the tangent line to the graph of f at x = 4 is y = 53x - 112.

Step-by-step explanation:

To find the derivative of f(x) = 7x^2 - 3x, we can use the power rule of differentiation. The power rule states that if we have a function f(x) = cx^n, where c is a constant and n is a real number, then the derivative of f(x) is given by f'(x) = cnx^(n-1).

Step 1:

Apply the power rule to find the derivative of 7x^2 - 3x.

f'(x) = 2(7)x^(2-1) - 1(3)x^(1-1)

= 14x - 3

Step 2:

Plug in the value x = 4 to find the slope at x = 4.

f'(4) = 14(4) - 3

= 56 - 3

= 53

Therefore, the slope of the tangent line at x = 4 is 53.

Using the point-slope form of a linear equation, we can find the equation of the tangent line.

Step 3:

Find the point on the graph of f(x) at x = 4.

f(4) = 7(4)^2 - 3(4)

= 112 - 12

= 100

Step 4:

Plug in the values m = 53 (slope) and (x, y) = (4, 100) (point on the line) into the point-slope form of a linear equation: y - y1 = m(x - x1).

y - 100 = 53(x - 4)

y - 100 = 53x - 212

y = 53x - 112

Therefore, the equation of the tangent line to the graph of f at x = 4 is y = 53x - 112.