Question

Find an equation of the tangent line to the curve y = 2ˣ at the given point.

Answer 1

To find the equation of the tangent line to the curve y = 2ˣ at a given point, determine the endpoints of the tangent and calculate the slope of the line. Then, plug in the coordinates of a point on the curve and the calculated slope into the point-slope form of the equation of a line.

Step-by-step explanation:

Solution:

1. Determine the endpoints of the tangent. These correspond to a position of 1300 m at time 19 s and a position of 3120 m at time 32 s.
2. Calculate the slope of the line using the formula: slope = (Y₂ - Y₁) / (X₂ - X₁), where Y₂ and Y₁ are the y-values of the endpoints, and X₂ and X₁ are the x-values of the endpoints.
3. Plug in the coordinates of a point on the curve (t = 25 s, y = 2ˣ) and the calculated slope into the point-slope form of the equation of a line: y - y₁ = m(x - x₁), where x₁, y₁ are the coordinates of the point and m is the slope.