Question

Find the equation of the tangent line to the graph of y = 4ˣ at the point where x=2.

a) y = 8x − 1
b) y = 16x − 8
c) y = 4x + 8
d) y = 32x − 16

Answer 1

To find the equation of the tangent line to the graph of y = 4ˣ at the point where x=2, we need to find the slope of the tangent line. The slope of the tangent line is equal to the derivative of the function y = 4ˣ.

Step-by-step explanation:

To find the equation of the tangent line to the graph of y = 4ˣ at the point where x=2, we need to find the slope of the tangent line.

The slope of the tangent line is equal to the derivative of the function y = 4ˣ. The derivative of 4ˣ is found using the power rule: d/dx (axⁿ) = naxⁿ⁻¹.

So, the derivative of y = 4ˣ is dy/dx = 4(2)ˣ⁻¹ = 4(2)¹ = 8.

Since the slope of the tangent line is 8, we can use the point-slope form of a linear equation to find the equation of the tangent line: y - y₁ = m(x - x₁), where (x₁, y₁) is the point of tangency and m is the slope.

Substituting the values x=2 and y=4ˣ into the equation, we get y - 4 = 8(x - 2).

Simplifying the equation, we get y = 8x - 16 + 4. So, the equation of the tangent line is y = 8x - 12.