Curve Sketching

Tangent to a Curve


Suppose y=f(x)y=f(x) is a curve such that the slope of the line tangent to the curve at an arbitrary point (x,y) (x, y) is given by xex.xe^{-x}. If y=f(x)y=f(x) passes through the origin (0,0),(0, 0), what is the value of f(18)?f(18)?

If the tangent lines to the curve y=2x3+ax2+bx+cy=2x^3+ax^2+bx+c at the two points (1,15)(1, 15) and (2,37)(2, 37) are parallel, what is a+b+c?a+b+c?

What is the equation of the tangent line to the curve y=5x28x+1y=5x^2-8x+1 at the point (4,49)?(4, 49)?

A curve y=ax2+bx+cy=ax^2+bx+c passes through the point (1,18)(1, 18) and the tangent line at the point (2,27)(2, 27) has slope 11. What is the sum of the constants a+b+c?a+b+c?

If the tangent line to the curve f(x)=x3+7 f(x) = x^{3}+7 at the point (4,71) (4, 71) is given by y=px+q,y=px+q, what is p+q?p+q?


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