Curve Sketching

Tangent to a Curve


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

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

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

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

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


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