ap-calculus-ab 1999 Q4

ap-calculus-ab · USA · free-response_formB Connected Rates of Change Parametric or Curve-Based Particle Motion Rates

(a) A point moves on the hyperbola $3 x ^ { 2 } - y ^ { 2 } = 23$ so that its $y$-coordinate is increasing at a constant rate of 4 units per second. How fast is the $x$-coordinate changing when $x = 4$ ? (b) For what values of $k$ will the line $2 x + 9 y + k = 0$ be normal to the hyperbola $3 x ^ { 2 } - y ^ { 2 } = 23$ ?

Suppose that the function $f$ has a continuous second derivative for all $x$, and that $f ( 0 ) = 2 , f ^ { \prime } ( 0 ) = - 3$, and $f ^ { \prime \prime } ( 0 ) = 0$. Let $g$ be a function whose derivative is given by $g ^ { \prime } ( x ) = e ^ { - 2 x } \left( 3 f ( x ) + 2 f ^ { \prime } ( x ) \right)$ for all $x$.

(a) A point moves on the hyperbola $3 x ^ { 2 } - y ^ { 2 } = 23$ so that its $y$-coordinate is increasing at a constant rate of 4 units per second. How fast is the $x$-coordinate changing when $x = 4$ ?
(b) For what values of $k$ will the line $2 x + 9 y + k = 0$ be normal to the hyperbola $3 x ^ { 2 } - y ^ { 2 } = 23$ ?

Paper Questions

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