Answer:
1) p = 14, p = -18
2) t = 14
Step-by-step explanation:
To find the values of p for which the line y = p - 9x is a tangent to the curve y = -x³ + 3x - 2, we need to find the point(s) of tangency where the two equations have the same value and slope.
The slope of the tangent line to y = -x³ + 3x - 2 at point (x, y) is the derivative of the function at that point.
Differentiate the function:
[tex]\dfrac{dy}{dx}=-3x^2+3[/tex]
The line y = p - 9x will be tangent to the curve when its slope equals the slope of the curve. The slope of y = p - 9x is -9, so set dy/dx equal to -9 and solve for x:
[tex]-3x^2+3=-9\\\\-3x^2=-12\\\\x^2=4\\\\x=\pm 2[/tex]
Therefore, the x-coordinates of the points of tangency are x = 2 and x = -2.
Find the corresponding y-coordinates by substituting x = 2 and x = -2 into the equation of the curve:
[tex]x=2 \implies y=-(2)^3+3(2)-2=-4\\\\x=-2 \implies y=-(-2)^3+3(-2)-2=0[/tex]
So, the points of tangency are (2, -4) and (-2, 0).
Substitute the points of tangency into the equation of the line y = p - 9x to find the corresponding values of p:
[tex]-4=p-9(2)\\\\-4=p-18\\\\p=14[/tex]
[tex]0=p-9(-2)\\\\0=p+18\\\\p=-18[/tex]
Therefore, the values of p when y = p - 9x is a tangent to y = -x³ + 3x - 2 are:
[tex]\Large\boxed{\boxed{y=14}}\\\\\Large\boxed{\boxed{y= -18}}[/tex]
[tex]\dotfill[/tex]
To find the value of t for which the line y + 2x = t is a tangent to the curve f(x) = 5 + 4x - x², we need to find the point of tangency where the two equations have the same value and slope.
The slope of the tangent line to f(x) = 5 + 4x - x² at point (x, y) is the derivative of the function at that point.
Differentiate the function:
[tex]f'(x)=4-2x[/tex]
The line y + 2x = t will be tangent to the curve when its slope equals the slope of the curve. The slope of y + 2x = t is -2, so set f'(x) equal to -2 and solve for x:
[tex]4-2x=-2\\\\-2x=-6\\\\x=3[/tex]
Therefore, the x-coordinate of the point of tangency is x = 3.
Find the corresponding y-coordinate by substituting x = 3 into the equation of the curve:
[tex]f(3)=5+4(3)-(3)^2\\\\f(3)=5+12-9\\\\f(3)=8[/tex]
So, the point of tangency is (3, 8)
Substitute the point of tangency into the equation of the line y + 2x = t to find the corresponding value of t:
[tex]8 + 2(3) = t\\\\t=14[/tex]
Therefore, the value of t when y + 2x = t is a tangent to f(x) = 5 + 4x - x² is:
[tex]\Large\boxed{\boxed{t=14}}[/tex]
