AP Calculus BC follows the curriculum set forth by the College Board. It is worth the equivalent of two semesters of college credit upon successful completion of the AP examination in May. Students enrolled in this course are required to take this test. Topics covered include library of functions, limits, the derivative, applications of the derivative, definite and indefinite integrals, applications of integration, techniques of integration, infinite series and sequences, etc.
notes / updates
February 24, 2025 | Feedback on today's quiz.
P1.
Be sure to draw and shade the region. When drawing the graph, give a little more detail (for example, the vertical asymptotes).
\( A = \int_2^{\infty} \frac{4}{x^2-1} dx = \lim\limits_{t\to \infty} \int_2^t \left( \frac{A}{x-1} + \frac{B}{x+1} \right) dx = ... = 2\ln 3 \)
Show work when calculating \( A = 2, B=-2 \).
P2.
Recognize the integral as improper due to the discontinuity. This means we NEED to use limits. Use u-substitution.
\( \int_{-4}^2 \frac{x}{\sqrt{16-x^2}} dx = \lim\limits_{b\to -4^+} \frac{-1}{2} \int_b^2 \frac{-2x}{\sqrt{16-x^2}} dx = ... = -2 \sqrt{3} \)
P3.
Rewrite the integral by completing the square:
\( \int \frac{1}{\sqrt{(2y-1)^2-4}} dy\)
Use trigonometric substitution with: \( 2y-1 = 2\sec\theta \).
Be sure to draw a right triangle. The final answer comes out to:
\( \frac{1}{2} \ln \left| \frac{2y-1+\sqrt{4y^2-4y-3}}{2} \right| + C\)
P4.
\( \int_0^1 \ x \sqrt{2-\sqrt{1-x^2}}dx\)
Option 1: Use u-substitution with \( u = 1-x^2 \), then w-substitution with \( w = 2-\sqrt{u}\).
Option 2: Use trigonometric substitution with \( x = \sin\theta \).
No matter which option you choose, be sure to update the bounds carefully whenever you switch variables.
The final answer is: \( \frac{2(8\sqrt{2}-7)}{15}\)
February 17, 2025 | Feedback on today's quiz.
P1.
\( \int \frac{1}{\sqrt{x^2-a^2}} dx \)
Use trigonometric substitution with: \( x = a sec\theta \).
Be sure to draw a right triangle with all three sides labeled.
\( dx = a \sec\theta \tan\theta d\theta \)
After the substitution of the expressions for \( x \) and \( dx \), the integral reduces to:
\( \int \sec\theta d\theta = \ln |\sec\theta + \tan\theta | + C = \ln\left| \frac{x +\sqrt{x^2-a^2}}{a} \right| + C \)
P2.
One of your classmates pointed out that I was missing a \( dx \) at the end of the integral for this problem.
\( \int_0^{3\sqrt{3}/2} \left( \frac{x^3}{(4x^2+9)^{3/2}} \right) dx \)
Option 1: \( u = 4x^2 +9, du = 8 x dx, x^3 = x^2 x = \frac{u-9}{4} x \)
Remember to switch the bounds of integration to \( 9 \leq u \leq 36 \) when switching from \( dx \) to \( du\).
Option 2: Use trigonometric substitution with \( x = \frac{3}{2} \tan\theta \)
Be sure to draw a right triangle and update the bounds: \( 0 \leq \theta \leq \frac{\pi}{3}\)
The final answer with either approach comes out to 3/32.
January 15, 2025 | Problem 59 (Section 4.1)
The initial velocity is \( v(0) = 16\) ft/s (positive because the object is moving up).
The initial position is \( s(0) = 64 \) ft.
a)
Find the velocity function: \( v(t) = -32 t + 16 \) by taking the anti-derivative of acceleration and using the initial value.
Find the position function \( s(t) = -16t^2 + 16t + 64 \) by taking the anti-derivative of velocity and using the initial value.
To find the time when the object hits the ground, set the position expression equal to zero, then solve.
b) Plug the t-value into the velocity function. January 13, 2025 | Feedback on Today's Quiz
P1.
We know for all four parts that \( f'(x) = v(x)\).
a) \( \int 2 v(x) dx = 2 \int v(x) dx = 2 f(x) + C \), so statement 1 is true.
b) The derivative of \( f(2x) \) is \( 2 f'(2x) = 2 v(2x) \), so the second statement is false.
\( f(x) = sin(x), v(x) = \cos(x) \) should work as an example.
c) \( \int (v(x) + 1) dx = f(x) + x + K \), so the third statement is false.
d) If we use the same example as the one in part b), we can see that \( (\sin x)^2 \) is not an anti-derivative of \( (\cos x)^2\). Therefore the fourth statement is false.
P2.
\( a(t) = -32 \rightarrow \int (-32) dt = -32t + C = v(t)\)
Since the initial velocity is zero, it follows that \(C = 0 \).
The velocity when the stone hits the ground is -120. The time at which this happens can be determined by solving:
\( v(t) = -32t = -120 \rightarrow t=\frac{15}{4} \).
the position function is: \( s(t) = \int v(t) dt = \int (-32 t) dt = -16t^2 + K \).
The position is zero when \( t = \frac{15}{4} \), so find \( K = 225\) feet.
The is also the height of the cliff since it is the value of the position function \( s(t) = -16t^2 + 225 \) when \( t =0 \)
P3.
a. \( \int \left[ \frac{1}{2\sqrt{1-x^2}} -\frac{3}{1+x^2} \right] dx = \frac{\arcsin x}{2} - 3\arctan x + C \)
b. \( \int \frac{\sec x + \cos x}{2 \cos x} dx = \int \left(\frac{1}{2}\sec^2x+\frac{1}{2} \right) dx = \frac{1}{2} \tan x + \frac{1}{2} x + C\)