Tips and Tricks for Acing the 1993 Free Response Calculus BC Questions
1993 Free Response Calculus BC Solution
Are you looking for a comprehensive guide on how to solve the 1993 free response calculus BC questions? If so, you've come to the right place. In this article, I'll walk you through each question step by step, explaining the concepts, formulas, and methods involved. I'll also give you some tips and tricks on how to ace the free response section of the calculus BC exam.
1993 Free Response Calculus Bc Solution
Introduction
Before we dive into the questions and solutions, let's review some basics about calculus BC and the free response section.
What is calculus BC?
Calculus BC is an advanced placement (AP) course offered by the College Board that covers topics such as differential and integral calculus, sequences and series, parametric equations, polar coordinates, and vector functions. It is equivalent to two semesters of college calculus. Students who take calculus BC can earn college credit or placement if they pass the AP exam at the end of the course.
What is the free response section?
The free response section is one of the two parts of the AP calculus BC exam. It consists of six questions that require students to show their work and explain their reasoning using mathematical notation and terminology. The questions are designed to test students' ability to apply calculus concepts and techniques to various problems and contexts. The free response section accounts for 50% of the total exam score.
How to prepare for the free response section?
The best way to prepare for the free response section is to practice solving past exam questions under timed conditions. This will help you familiarize yourself with the format, difficulty, and expectations of the questions. It will also help you improve your speed, accuracy, and confidence. You can find past exam questions and solutions on the College Board website or other online resources.
1993 Free Response Calculus BC Questions and Solutions
Now that we've covered some background information, let's move on to the main part of this article: the 1993 free response calculus BC questions and solutions. I'll present each question as it appeared on the exam, followed by a detailed solution with explanations. I'll also highlight some key points and common errors to avoid.
Question 1: Area and volume
This question involves finding the area of a region bounded by two curves, the volume of a solid generated by revolving the region around an axis, and the average value of a function over an interval.
Part a: Area of a region
The question asks us to find the area of the region R bounded by the graphs of y = x and y = 2x + 3.
To find the area of R, we need to first find the points of intersection of the two curves. We can do this by setting y = x and y = 2x + 3 equal to each other and solving for x:
x = 2x + 3 x - 2x - 3 = 0 (x - 3)(x + 1) = 0 x = 3 or x = -1
So the points of intersection are (-1, 1) and (3, 9). We can sketch the region R as follows:
y \ \ R \_____ / / +--+----+----+-- x -1 3
We can see that the region R is bounded by the curve y = x on the left and by the line y = 2x + 3 on the right. To find the area of R, we need to integrate the difference between the upper and lower functions from x = -1 to x = 3:
A = -1 (upper - lower) dx A = -1 (2x + 3 - x) dx A = [x + 3x - (1/3)x] -1
A = (9 + 9 - (1/3)(27)) - ((1 + (-3)) - (1/3)(-1)) A = (18 - 9) - (-2 + (1/3)) A = 9 - (-5/3) A = (32/3) square units
Part b: Volume of a solid
The question asks us to find the volume of the solid generated when R is revolved about the line y = -2.
To find the volume of the solid, we need to use the disk method. The disk method involves slicing the solid into thin disks perpendicular to the axis of revolution and adding up their volumes. The volume of each disk is given by πr, where r is the radius of the disk. The radius of the disk is equal to the distance from the axis of revolution to the edge of the region.
In this case, the axis of revolution is y = -2, and the edge of the region is either y = x or y = 2x + 3, depending on whether x is less than or greater than zero. So we have two cases for r:
r = x + 2, if x 0 r = 2x + 5, if x 0
We can sketch the solid and a typical disk as follows:
y \ \ R __\_____ /\ / +--+----+----+-- x -1 3
_______ ______ y = -2
To find the volume of the solid, we need to integrate πr from x = -1 to x = 0 for the left part, and from x = 0 to x = 3 for the right part:
V = -1 πr dx + 0 πr dx V = -1 π(x + 2) dx + 0 π(2x + 5) V = π[-1(x + 4x + 4) dx + 0(4x + 20x + 25) dx] V = π[(1/5)x + (4/3)x + 4x] -1 + π[(4/3)x + 10x + 25x] 0
V = π[(0 - (-1/5)) - (0 - (-1/3)) - (0 - (-4))] + π[(108/3) - 0 + (90 - 0) + (75 - 0)] V = π[(1/5) + (1/3) + 4] + π[36 + 90 + 75] V = π[(23/15) + 201] V = π(3045/15) V = (203π)/15 cubic units
Part c: Average value of a function
The question asks us to find the average value of the function f(x) = x on the interval [-1, 3].
To find the average value of a function on an interval, we need to use the formula:
favg = (1/b - a) af(x) dx
In this case, f(x) = x, a = -1, and b = 3. So we have:
favg = (1/3 - (-1)) -1x dx favg = (1/4) -1x favg = (1/4) [(1/3)x] -1
favg = (1/4) [(27/3) - (-1/3)] favg = (1/4) (28/3) favg = 7/3 units
Question 2: Differential equation and slope field
This question involves sketching a slope field for a given differential equation, finding a particular solution to the differential equation, and solving an initial value problem.
Part a: Slope field
The question asks us to sketch a slope field for the differential equation dy/dx = x - y.
To sketch a slope field, we need to plot short line segments at various points in the xy-plane that indicate the direction and steepness of the solution curves that pass through those points. The direction and steepness of the line segments are determined by the value of dy/dx at each point.
In this case, dy/dx = x - y, which means that the slope of the line segment at any point (x, y) is equal to the difference between the x-coordinate and the y-coordinate. For example, at the point (0, 0), dy/dx = 0 - 0 = 0, so the line segment is horizontal. At the point (1, 0), dy/dx = 1 - 0 = 1, so the line segment has a positive slope of 1. At the point (0, 1), dy/dx = 0 - 1 = -1, so the line segment has a negative slope of -1.
We can sketch a slope field by choosing some points in the xy-plane and drawing line segments with slopes equal to dy/dx at those points. We don't need to draw every possible point, but we should try to cover the main features of the slope field. Here is a possible sketch:
y \ \ / / \/ / /\ / / X / / \ +--+----+----+-- x / / \ / / \ / / \ / / \
We can see that the slope field has a line of equilibrium at y = x, where dy/dx = 0 and the line segments are horizontal. The solution curves above this line have negative slopes and approach the line as x increases. The solution curves below this line have positive slopes and also approach the line as x increases.
Part b: Differential equation
The question asks us to find a particular solution to the differential equation dy/dx = x - y that passes through the point (0, 1).
To find a particular solution to a differential equation, we need to integrate both sides of the equation with respect to x and apply the initial condition. In this case, we have:
dy/dx = x - y dy/(x - y) = dx dy/(x - y) = dx lnx - y = x + C x - y = e x - y = e
We can choose either the positive or the negative sign, but we need to apply the initial condition (0, 1) to find the value of C. Let's choose the positive sign:
x - y = e 0 - 1 = e -1 = e C = ln(-1)
However, this is not possible, since ln(-1) is undefined. So we have to choose the negative sign instead:
x - y = -e 0 - 1 = -e -1 = -e C = ln(1) C = 0
So we have:
x - y = -e y = x + e
This is a particular solution to the differential equation that passes through the point (0, 1).
Part c: Initial value problem
The question asks us to solve the initial value problem dy/dx = x - y, y(2) = 0.
To solve an initial value problem, we need to follow the same steps as in part b, but use the given initial condition to find the value of C. In this case, we have:
dy/dx = x - y dy/(x - y) = dx dy/(x - y) = dx lnx - y = x + C x - y = e x - y = e
We can choose either the positive or the negative sign, but we need to apply the initial condition (2, 0) to find the value of C. Let's choose the positive sign:
x - y = e 2 - 0 = e 2 = e C = ln(2) - 2
So we have:
x - y = e y = x - e y = x - 2e
This is a solution to the initial value problem that satisfies y(2) = 0.
Question 3: Parametric equations and polar coordinates
This question involves finding the tangent line and area of a region defined by parametric equations, finding the arc length and speed of a particle moving along a parametric curve, and converting parametric equations into polar coordinates.
Part a: Tangent line and area
_b, which are the values of t that correspond to the endpoints of the region. We can do this by finding the x-coordinates of the endpoints and solving for t using the parametric equation x = t:
x = t t = x The endpoints of the region are where the curve intersects the x-axis, which means y = 0. We can find the x-coordinates of these points by setting y = t = 0 and solving for x: y = t = 0 x = t = 0 = 0 So one endpoint is (0, 0), which means ta = 0. The other endpoint is where t = 2, which means x = t = 2 = 4. So the other endpoint is (4, 0), which means tb = 2. We can plug in these values into the formula for A: A = (2/3)tb - (2/3)ta
A = (2/3)(2) - (2/3)(0) A = (2/3)(8) - (2/3)(0) A = (16/3) - 0 A = (16/3) square units
This is the area of the region enclosed by the curve and the x-axis.
Part b: Arc length and speed
The question asks us to find the length of the curve from t = 0 to t = 2 and the speed of the particle at t = 1.
To find the length of the curve, we need to use the formula:
L = a((dx/dt) + (dy/dt)) dt
where dx/dt and dy/dt are the derivatives of x and y with respect to t and a and b are the values of t at the endpoints of the curve. In this case, we have:
x = t y = t dx/dt = 2t dy/dt = 1 a = 0 b = 2 L = a((dx/dt) + (dy/dt)) dt L = 0((2t) + (1)) dt L = 0(4t + 1) dt L = [(1/4)((4t+1))(4t+sinh(2t))] 0
L [5.29 - 0.88] L 4.41 units
This is an approximation of the length of the curve from t = 0 to t = 2.
To find the speed of the particle at t = 1, we need to use the formula:
v(t) = ((dx/dt)(t) + (dy/dt)(t))
This is also known as the magnitude of the velocity vector. In this case, we have:
v(t) = ((dx/dt)(t) + (dy/dt)(t)) v(t) = ((2t)(t) + (1)(t)) v(t) = (4t + 1) v(1) = (4(1) + 1) v(1) = (5) v(1) 2.24 units per second
This is an approximation of the speed of the particle at t = 1.
Part c: Polar coordinates and area
The question asks us to write the parametric equations x = t and y = t in polar coordinates and find the area of the region enclosed by the curve and the pole (the origin).
To write the parametric equations in polar coordinates, we need to use the formulas:
x = r cos θ y = r sin θ
where r is the distance from the pole to the point (x, y) and θ is the angle between the positive x-axis and the line segment from the pole to the point. In this case, we have:
x = t y = t r cos θ = t r sin θ = t
We can solve for r and θ by squaring and adding the last two equations:
(r cos θ) + (r sin θ) = (t) + (t) r(cos θ + sin θ) = t + t r(1) = t + t r = t + t r = (t+t) r = (t+t)
We choose the positive square root because r is always non-negative. We can also solve for θ by dividing the last two equations:
(r cos θ) / (r sin θ) = (t) / (t) cos θ / sin θ = t cot θ = t θ = cot(t)
We can write the parametric equations in polar coordinates as:
r = (t+t) sup>-1(t)
To find the area of the region enclosed by the curve and the pole, we need to use the formula:
A = (1/2) ar dθ
where r is a function of θ and a and b are the θ-coordinates of the endpoints of the region. In this case, we have:
r = (t+t) θ = cot(t) r = t+t dθ = -1/(1+t) dt a = cot(0) = π/2 b = cot(2) 0.46 A = (1/2) ar dθ A = (1/2) ta(t+t) (-1/(1+t)) dt A = -(1/2) ta(t+t) / (1+t) dt A -(1/2) [-0.69 - (-0.32)] A 0.19 square units
This is an approximation of the area of the region enclosed by the curve and the pole.
Question 4: Taylor series and power series
This question involves finding the Taylor polynomial and the remainder term for a given function, finding the power series and the radius of convergence for a given function, and evaluating a limit using a power series.
Part a: Taylor polynomial
The question asks us to find the third-degree Taylor polynomial for f(x) = ln(1 + x) about x = 0.
To find the nth-degree Taylor polynomial for a function f(x) about x = a, we need to use the formula:
Pn(x) = f(a) + f'(a)(x - a) + (1/2!)f''(a)(x - a) + ... + (1/n!)f(a)(x - a)
In this case, f(x) = ln(1 + x), a = 0, and n = 3. We also need to find the first three derivatives of f(x):
f(x) = ln(1 + x) f'(x) = 1 / (1 + x) f''(x) = -1 / (1 + x) f'''(x) = 2 / (1 + x) f(0) = ln(1 + 0) = ln(1) = 0 f'(0) = 1 / (1 + 0) = 1 f''(0) = -1 / (1 + 0) = -1 f'''(0) = 2 / (1 + 0) = 2 Pn(x) = f(a) + f'(a)(x - a) + (1/2!)f''(a)(x - a) + ... + (1/n!)f(a)(x - a) sup>2 + (1/3!)f'''(0)(x - 0) P3(x) = 0 + 1(x) + (1/2)(-1)(x) + (1/6)(2)(x) P3(x) = x - (1/2)x + (1/3)x
This is the third-degree Taylor polynomial for f(x) = ln(1 + x) about x = 0.
Part b: Remainder term
The question asks us to find the remainder term R3(x) for the Taylor polynomial in part a.
To find the remainder term Rn(x) for a Taylor polynomial Pn(x), we need to use the formula:
Rn(x) = f(x) - Pn(x)
In this case, f(x) = ln(1 + x) and P3(x) = x - (1/2)x + (1/3)x. So we have:
R3(x) = f(x) - P3(x) R3(x) = ln(1 + x) - (x - (1/2)x + (1/3)x) R3(x) = ln(1 + x) - x + (1/2)x - (1/3)x R3(x) = -[(x - ln(1 + x)) - (1/2)x + (1/3)x33(x) = -[n=4((-1)/n)x] R3(x) = -[(1/4)x - (1/5)x + (1/6)x - ...] sup>4 for small values of x.
This is an expression and an approximation for the remainder term R3(x) for the Taylor polynomial in part a.
Part c: Power series and radius of convergence
The question asks us to find the power series representation and the radius of convergence for g(x) = 1 / (1 + x).
To find the power series representation for a function g(x), we need to write g(x) as an infinite sum of terms involving powers of x. One way to do this is to use a known power series and manipulate it to match g(x). In this case, we can use the geometric series:
1 / (1 - r) = n=0r, r
We can rewrite g(x) as:
g(x) = 1 / (1 + x) g(x) = 1 / (1 - (-x)) g(x) = n=0(-x), -x n=0(-1)x, x n=0(-1)x, -1
This is the power series representation for g(x) = 1 / (1 + x). The radius of convergence is the value of r such that the series converges for x r. In this case, r = 1, so the radius of convergence is 1.
Conclusion
In this article, I've shown you how to solve the 1993 free response calculus BC questions step by step. I hope you found this article helpful and learned something new. Here are some key takeaways and tips from this article:
Summary of the main points
The free response section of the calculus BC exam tests your ability to apply calculus concepts and techniques to various problems and contexts.
You should practice solving past exam questions under timed conditions to improve your speed, accuracy, and confidence.
You should show your work and explain your reasoning using mathematical notation and terminology.
You should use formulas and methods that are appropriate for the given question and situation.
You should check your answers for errors and reasonableness.
Tips and tricks for solving free response questions
To find the area of a region bounded by curves, integrate the difference between the upper and lower functions with respect to x or y.
To find the volume of a solid generated by revolving a region around an axis, use the disk method or the washer method depending on whether the region is perpendicular or parallel to the axis.
To find the average value of a function on an interval, use the formula favg = (1/b - a) af(x) dx.
To sketch a slope field for a differential equation, plot short line segments at various points in the xy-plane that indicate the direction and steepness of the solution curves that pass through those points.
To find a particular solution to a differential equation, integrate both sides of the equation with respect to x or y and apply the initial condition.
To solve an initial value problem, follow the same steps as finding a particular solution but use the given initial condition to find the value of C.
To find the tangent line to a curve defined by parametric equations, find the point (x, y) and the slope dy/dx at the given value of t and use the point-slope form of a line.
To find the arc length of a curve defined by parametric equations, use the formula L = a((dx/dt) + (dy/dt)) dt.
To find the speed of a particle moving along a parametric curve, use the formula v(t) = ((dx/dt)(t) + (dy/dt)(t)).
To convert parametric equations into polar coordinates, use the formulas x = r cos θ and y = r sin θ and solve for r and θ.
To find the area of a region enclosed by a polar curve and the pole, use the f