Determine the slope-intercept form of the equation of the tangent line to y = 2sqrt(2) + picos(x) at the point,
x = pi/4
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To find the slope-intercept form of the equation of the tangent line, we need to find the slope and the y-intercept.
Step 1: Find the derivative of the given function.
Taking the derivative of y = 2√2 + πcos(x) with respect to x gives us:
dy/dx = -πsin(x)
Step 2: Evaluate the derivative at the given point.
To find the slope of the tangent line at x = π/4, substitute x = π/4 into the derivative:
dy/dx = -πsin(π/4) = -π/√2 = -π√2/2
Step 3: Find the y-coordinate of the given point.
Substitute x = π/4 into the original function:
y = 2√2 + πcos(π/4) = 2√2 + π/√2 = 2√2 + π√2/2
Step 4: Use the slope-intercept form of the equation (y = mx + b) and plug in the values from Steps 2 and 3.
Slope (m) = -π√2/2
y-intercept (b) = 2√2 + π√2/2
Therefore, the slope-intercept form of the equation of the tangent line is:
y = (-π√2/2)x + (2√2 + π√2/2)
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