consider the function y = 2sinx
a) determine dy/dx
b) calculate the exact value of x for which the slope of the tangent to the graph is √2 in the interval [0, pie]
c) determine the equation of the tangent in part (b).
d) determine the y-intercept of the tangent in part (c).
a) To determine dy/dx, we need to differentiate the function y = 2sin(x) with respect to x.
The derivative of sin(x) is cos(x), and when we take the derivative of 2sin(x), the constant 2 remains unchanged. Therefore, dy/dx is given by:
dy/dx = 2cos(x)
b) To find the x-value where the slope of the tangent to the graph is √2, we need to set dy/dx equal to √2 and solve for x in the interval [0, π].
2cos(x) = √2
Divide both sides by 2:
cos(x) = √2 / 2
To find the x-values in the interval [0, π] that satisfy this equation, we can use the inverse cosine function (cos^-1) or the unit circle. Here, we know that cos(π/4) = √2 / 2, so x = π/4 is a solution in the given interval.
c) To determine the equation of the tangent line at the point (x, y) where x = π/4, we need to find the y-coordinate (which is y = 2sin(x)) and the slope (which is given by dy/dx).
At x = π/4, y = 2sin(π/4) = 2(√2 / 2) = √2.
Therefore, the tangent line has the point (π/4, √2) and a slope of √2.
We can use the point-slope form of a line to find the equation of the tangent:
y - √2 = √2(x - π/4)
Simplifying:
y - √2 = √2x - π/2
y = √2x - π/2 + √2
So, the equation of the tangent line is y = √2x - π/2 + √2.
d) To find the y-intercept of the tangent line in part (c), we set x = 0 in the equation y = √2x - π/2 + √2 and solve for y.
y = √2(0) - π/2 + √2
y = -π/2 + √2
Therefore, the y-intercept of the tangent line is -π/2 + √2.