ap-calculus-bc

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2000 free-response

5 maths questions

Q2 Variable acceleration (1D) Two-particle comparison problem View
2. Two runners, $A$ and $B$, run on a straight racetrack for $0 \leq t \leq 10$ seconds. The graph above, which consists of two line segments, shows the velocity, in meters per second, of Runner $A$. The velocity, in meters per second, of Runner $B$ is given by the function $v$ defined by $v ( t ) = \frac { 24 t } { 2 t + 3 }$.
(a) Find the velocity of Runner $A$ and the velocity of Runner $B$ at time $t = 2$ seconds. Indicate units of measure.
(b) Find the acceleration of Runner $A$ and the acceleration of Runner $B$ at time $t = 2$ seconds. Indicate units of measure.
(c) Find the total distance run by Runner $A$ and the total distance run by Runner $B$ over the time interval $0 \leq t \leq 10$ seconds. Indicate units of measure.
Q3 Taylor series Construct Taylor/Maclaurin polynomial from derivative values View
3. The Taylor series about $x = 5$ for a certain function $f$ converges to $f ( x )$ for all $x$ in the interval of convergence. The $n$th derivative of $f$ at $x = 5$ is given by $f ^ { ( n ) } ( 5 ) = \frac { ( - 1 ) ^ { n } n ! } { 2 ^ { n } ( n + 2 ) }$, and $f ( 5 ) = \frac { 1 } { 2 }$.
(a) Write the third-degree Taylor polynomial for $f$ about $x = 5$.
(b) Find the radius of convergence of the Taylor series for $f$ about $x = 5$.
(c) Show that the sixth-degree Taylor polynomial for $f$ about $x = 5$ approximates $f ( 6 )$ with error less than $\frac { 1 } { 1000 }$.
END OF PART A OF SECTION II
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CALCULUS BC SECTION II, Part B Time-45 minutes Number of problems-3 No calculator is allowed for these problems.
Q4 Variable acceleration (vectors) View
4. A moving particle has position $( x ( t ) , y ( t ) )$ at time $t$. The position of the particle at time $t = 1$ is $( 2,6 )$, and the velocity vector at any time $t > 0$ is given by $\left( 1 - \frac { 1 } { t ^ { 2 } } , 2 + \frac { 1 } { t ^ { 2 } } \right)$.
(a) Find the acceleration vector at time $t = 3$.
(b) Find the position of the particle at time $t = 3$.
(c) For what time $t > 0$ does the line tangent to the path of the particle at $( x ( t ) , y ( t ) )$ have a slope of 8 ?
(d) The particle approaches a line as $t \rightarrow \infty$. Find the slope of this line. Show the work that leads to your conclusion.
Q5 Implicit equations and differentiation Verify implicit derivative and find tangent line features View
5. Consider the curve given by $x y ^ { 2 } - x ^ { 3 } y = 6$.
(a) Show that $\frac { d y } { d x } = \frac { 3 x ^ { 2 } y - y ^ { 2 } } { 2 x y - x ^ { 3 } }$.
(b) Find all points on the curve whose $x$-coordinate is 1 , and write an equation for the tangent line at each of these points.
(c) Find the $x$-coordinate of each point on the curve where the tangent line is vertical.
Q6 Differential equations Multi-Part DE Problem (Slope Field + Solve + Analyze) View
6. Consider the differential equation given by $\frac { d y } { d x } = x ( y - 1 ) ^ { 2 }$.
(a) On the axes provided, sketch a slope field for the given differential equation at the eleven points indicated. (Note: Use the axes provided in the pink test booklet.) [Figure]
(b) Use the slope field for the given differential equation to explain why a solution could not have the graph shown below. [Figure]
(c) Find the particular solution $y = f ( x )$ to the given differential equation with the initial condition $f ( 0 ) = - 1$.
(d) Find the range of the solution found in part (c).
END OF EXAMINATION
Copyright © 2000 College Entrance Examination Board and Educational Testing Service. All rights reserved. AP is a registered trademark of the College Entrance Examination Board.