find the equation of tangent line to the graph of f(x)=xcos(√2 x) at a=0
To find the equation of the tangent line to the graph of f(x) = xcos(√2 x) at a = 0, we need to determine the slope of the tangent line and a point on the line.
Step 1: Calculate the derivative of f(x) using the product rule and the chain rule.
f'(x) = cos(√2 x) - √2xsin(√2 x)
Step 2: Determine the slope of the tangent line by evaluating f'(x) at x = 0.
f'(0) = cos(√2 * 0) - √2 * 0 * sin(√2 * 0)
= cos(0)
= 1
Therefore, the slope of the tangent line at x = 0 is 1.
Step 3: Find a point on the line. Since we are interested in the tangent line at x = 0, we can find the corresponding y-value by evaluating f(x) at x = 0.
f(0) = 0 * cos(√2 * 0)
= 0
Therefore, the point (0, 0) lies on the tangent line.
Step 4: Write the equation of the tangent line using the slope and a point.
Using the point-slope form of a linear equation, we have:
y - y₁ = m(x - x₁),
where m is the slope and (x₁, y₁) is a point on the line.
Plugging in the values we have:
y - 0 = 1(x - 0)
y = x
Therefore, the equation of the tangent line to the graph of f(x) = xcose(√2 x) at x = 0 is y = x.