Calculus

# 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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