AP Score Calculator

AP Calculus BC Score Calculator 2025 | Instant AP Calc BC Composite

3 weeks ago
5 mins
73 Views

AP® Calculus BC Score Calculator

∫

Adjust the sliders below to calculate your potential AP® score.

Section I: Multiple-Choice 0/45
Section II: Free Response Questions
Question 10/9
Question 20/9
Question 30/9
Question 40/9
FRQ 5 (AB Part)0/5
FRQ 5 (BC Part)0/4
Question 60/9
Your Estimated AP® Score
1
Keep practicing those calculus concepts!
MCQ Score (scaled to 54)
0
FRQ Score (out of 54)
0
Total Composite Score
0/108
020406080108
Score Thresholds (approximate):
1 (0-34) • 2 (35-40) • 3 (41-51) • 4 (52-61) • 5 (62+)
Official AP Calculus BC Practice

AP Calculus BC – 2025 Cheatsheet

👑 Unit 1: Limits & Continuity ▼
  • Order of growth rates from fastest to slowest: \(x^x, x!, a^x, x^n, x\ln x, \ln x\)
  • Methods to algebraically simplify limits if you can't directly plug in: Completing the square, Rationalization, Factoring
  • Limit Properties:
    • \(\lim_{x\to c}(af(x)) = a\lim_{x\to c}f(x)\)
    • \(\lim_{x\to c}(f(x) \pm g(x)) = \lim_{x\to c}f(x) \pm \lim_{x\to c}g(x)\)
    • \(\lim_{x\to c}(f(x)g(x)) = \lim_{x\to c}f(x) \cdot \lim_{x\to c}g(x)\)
    • \(\lim_{x\to c}\frac{f(x)}{g(x)} = \frac{\lim_{x\to c}f(x)}{\lim_{x\to c}g(x)}\), provided \(\lim_{x\to c}g(x) \neq 0\)
    • \(\lim_{x\to c}f(g(x)) = f(\lim_{x\to c}g(x))\), if f is continuous at \(\lim_{x\to c}g(x)\)
  • Continuity exists if \(f(c) = \lim_{x\to c}f(x)\)
  • Intermediate Value Theorem: Write "Since f(x) is continuous on [a,b] and f(c) is between f(a) and f(b), by the IVT there is a c in (a,b) such that f(c)=0"
🤓 Unit 2: Fundamentals of Differentiation ▼
  • All differentiable functions are continuous, but not all continuous functions are differentiable
  • Average Rate of Change = \(\frac{f(x+h)-f(x)}{h}\) or \(\frac{f(b)-f(a)}{b-a}\)
  • Derivative Definition: \(f'(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}{h} = \lim_{z\to x}\frac{f(z)-f(x)}{z-x}\)
  • Power Rule: \(\frac{d}{dx}x^n = nx^{n-1}\)
  • Sum/Difference Rule: \(\frac{d}{dx}[f(x)\pm g(x)] = f'(x)\pm g'(x)\)
  • Product Rule: \(\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)\)
  • Quotient Rule: \(\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x)-f(x)g'(x)}{[g(x)]^2}\)
  • Common Derivatives:
    • \(\frac{d}{dx}c = 0\) (c is constant)
    • \(\frac{d}{dx}e^x = e^x\)
    • \(\frac{d}{dx}kf(x) = kf'(x)\)
    • \(\frac{d}{dx}\ln(x) = \frac{1}{x}\)
    • \(\frac{d}{dx}\sin(x) = \cos(x)\)
    • \(\frac{d}{dx}\cos(x) = -\sin(x)\)
    • \(\frac{d}{dx}\tan(x) = \sec^2(x)\)
    • \(\frac{d}{dx}\cot(x) = -\csc^2(x)\)
    • \(\frac{d}{dx}\sec(x) = \sec(x)\tan(x)\)
    • \(\frac{d}{dx}\csc(x) = -\csc(x)\cot(x)\)
📈 Unit 3: Composite, Implicit, & Inverse Functions ▼
  • Chain Rule: \(\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)\)
  • \(\frac{d}{dx}a^x = a^x\ln(a)\)
  • \(\frac{d}{dx}\log_a(x) = \frac{1}{x\ln(a)}\)
  • Implicit Differentiation: Differentiate each term with respect to the individual variables, multiply by \(\frac{dy}{dx}\) when differentiating terms with y
  • Derivatives of Inverse Trig Functions:
    FunctionDerivative
    \(\sin^{-1}x\)\(\frac{1}{\sqrt{1-x^2}}\)
    \(\cos^{-1}x\)\(\frac{-1}{\sqrt{1-x^2}}\)
    \(\tan^{-1}x\)\(\frac{1}{1+x^2}\)
    \(\cot^{-1}x\)\(\frac{-1}{1+x^2}\)
    \(\sec^{-1}x\)\(\frac{1}{|x|\sqrt{x^2-1}}\)
    \(\csc^{-1}x\)\(\frac{-1}{|x|\sqrt{x^2-1}}\)
👀 Unit 4: Contextual Applications of Differentiation ▼
  • Particle Motion:
    • Position = \(s(t)\)
    • Velocity = \(v(t) = s'(t)\)
    • Acceleration = \(a(t) = v'(t) = s''(t)\)
  • If velocity is negative, the particle is moving to the left.
  • If velocity is positive, the particle is moving to the right.
  • If velocity and acceleration have the same sign, the particle is speeding up.
  • If velocity and acceleration have different signs, the particle is slowing down.
  • Steps for Related Rates:
    1. Draw a picture and label the picture (assign variables)
    2. List your knowns and unknown values
    3. Write an equation to model the situation
    4. Take the derivative of both sides. Remember: \(\frac{d}{dt}\)
    5. Plug in known values and solve for desired values. DON'T FORGET UNITS!
  • Linearization: \(f(c+a) \approx f(c) + f'(c)a\)
  • L'Hopital's Rule: If \(\lim_{x\to c}\frac{f(x)}{g(x)}\) is indeterminate (\(0/0\) or \(\infty/\infty\)), then \(\lim_{x\to c}\frac{f(x)}{g(x)} = \lim_{x\to c}\frac{f'(x)}{g'(x)}\)
✨ Unit 5: Analytical Applications of Differentiation ▼
  • Mean Value Theorem: Write "Since f(x) is continuous on [a,b] and differentiable on (a,b), there exists a c in (a,b) such that \(f'(c) = \frac{f(b)-f(a)}{b-a}\) by the MVT."
  • Extreme Value Theorem: Write "Since f(x) is continuous on [a,b], by the EVT, there exists at least one local maximum and one local minimum on [a,b]."
  • Critical Points: where \(f'(x) = 0\) or does not exist
  • \(f'(x) > 0\): increasing; \(f'(x) < 0\): decreasing
  • First Derivative Test: where \(f'(x) = 0\), if \(f'(x)\) changes from + to −: local max; if \(f'(x)\) changes from − to +: local min
  • Determining Concavity: \(f''(x) > 0\): concave up; \(f''(x) < 0\): concave down; \(f''(x) = 0\): potential inflection point
  • Second Derivative Test: If \(f'(x) = 0\), then:
    • if \(f''(x) > 0\): local min
    • if \(f''(x) < 0\): local max
    • if \(f''(x) = 0\): test is inconclusive
  • Steps for Optimization:
    1. Draw picture
    2. Label your picture and assign variables
    3. Write an equation and use given information to find relationships among variables
    4. Find extrema (min/max) and evaluate the function
🔥 Unit 6: Integration of Accumulation of Change ▼
  • The integral is the area between the graph and the x-axis
  • Riemann Sum can be used to approximate area (includes left, right, midpoint, and trapezoidal sum)
    • \(\int_a^b f(x)dx \approx \lim_{n\to\infty} \sum_{i=1}^n f(x_i^*)\Delta x\) where \(\Delta x = \frac{b-a}{n}\) and \(x_i = a + i\Delta x\)
  • Fundamental Theorem of Calculus (FTC): If \(F'(x) = f(x)\), then \(\int_a^b f(x)dx = F(b) - F(a)\)
  • Basic Integration Formulas (NEVER FORGET + C):
    • \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\) (n ≠ -1)
    • \(\int \frac{1}{x} dx = \ln|x| + C\)
    • \(\int e^x dx = e^x + C\)
    • \(\int \sin(x) dx = -\cos(x) + C\)
    • \(\int \cos(x) dx = \sin(x) + C\)
    • \(\int \sec^2(x) dx = \tan(x) + C\)
    • \(\int \csc^2(x) dx = -\cot(x) + C\)
    • \(\int \sec(x)\tan(x) dx = \sec(x) + C\)
    • \(\int \csc(x)\cot(x) dx = -\csc(x) + C\)
  • Integration by Parts (IBP): \(\int u\,dv = uv - \int v\,du\)
  • Learn the Tabular Method to make IBP easier
  • Use Partial Fraction Decomposition to integrate rational functions
  • Improper Integrals: \(\int_a^{\infty} f(x)dx = \lim_{b\to\infty} \int_a^b f(x)dx\)
💎 Unit 7: Differential Equations ▼
  • A slope field is a graphical representation of a differential equation in the form \(dy/dx = f(x,y)\)
  • Logistic Differential Equation: \(\frac{dP}{dt} = kP\left(1 - \frac{P}{L}\right)\) where P is the population, L is the carrying capacity, and k is a constant
  • Euler's method can be used to find the numerical values of functions based on a given differential equation and an initial condition
🐶 Unit 8: Applications of Integration ▼
  • Average Value: \(\frac{1}{b-a} \int_a^b f(x)dx\)
  • Average Value Theorem: Write "Since f(x) is continuous on [a,b], by the AVT, there must be a c in (a,b) where \(f(c) = \frac{1}{b-a} \int_a^b f(x)dx\)."
  • Position and velocity relationship:
    • Acceleration = \(a(t)\)
    • Velocity = \(v(t) = \int a(t) dt\)
    • Position = \(s(t) = \int v(t) dt\)
    • Speed = \(|v(t)|\)
    • Distance traveled = \(\int |v(t)| dt\)
  • Volume using the Washer Method: \(\pi \int (R_{\text{outer}}^2 - R_{\text{inner}}^2) dx\)
  • Volume using the Disc Method: \(\pi \int R^2 dx\)
  • Arc Length: \(\int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2}dx\)
🦖 Unit 9: Parametric, Polar & Vector Functions ▼
  • Second Derivative of Parametric Equation: \(\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}\)
  • Arc Length for Parametric Functions: \(s = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt\)
  • Slope of a Tangent Line for Polar Equations: \(\frac{dy}{dx} = \frac{\frac{dr}{d\theta}\sin\theta + r\cos\theta}{\frac{dr}{d\theta}\cos\theta - r\sin\theta}\)
  • Polar Conversions: \(x = r \cos\theta\), \(y = r \sin\theta\), \(r = \sqrt{x^2 + y^2}\)
  • Area under Polar Curves: \(A = \int_\alpha^\beta \frac{1}{2}r^2 d\theta\)
  • Area under two Polar Curves: \(A = \int_\alpha^\beta \frac{1}{2}(r_1^2 - r_2^2) d\theta\)
  • Arc Length for Polar Functions: \(L = \int_\alpha^\beta \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} d\theta\)
♾️ Unit 10: Infinite Sequences & Series ▼
  • Sequences: arithmetic and geometric
  • Series:
    • Harmonic series \(\sum \frac{1}{n}\) diverges
    • Power Series with terms \(\frac{1}{n^p}\) converges when \(p > 1\), else it diverges
    • Alternating Series: For an alternating series (terms change sign), converges if \(\lim_{n\to\infty} a_n = 0\), else it diverges
  • Taylor Series: \(f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!}(x-c)^n\)
  • Maclaurin Series (make sure you memorize them!): Taylor series centered at \(c = 0\)
  • Various tests used to determine convergence and divergence:
    • nth Term Test: If \(\lim_{n\to\infty} a_n \neq 0\), then the series diverges
    • Limit Comparison Test
    • Direct Comparison Test
    • Integral Test
    • Alternating Series Test
    • Ratio Test
💯 FRQ Tips & Strategies ▼
  • Be sure to show ALL of your work (even when using a calculator).
  • Work on all of the parts that you know first before moving onto other parts. (Get those easy points!)
  • Do not round any values as you complete the problem! Wait all the way until the end to round your answer to 3 or 4 decimal places.
  • You can use abbreviations like IVT (Intermediate Value Theorem), MVT (Mean Value Theorem), and FTOC (Fundamental Theorem of Calculus)
  • Do not simplify your answers unless specified. You don't want to lose points on steps you don't need to do!
  • Memorize your important theorems and convergence tests! You'll need to know the conditions where the theorems and tests are met.
  • Keep an eye on the time and pace yourself.
  • For problems involving derivatives, clearly identify the rule you're using (chain rule, product rule, etc.)
  • For series problems, clearly state which convergence test you're applying and why
  • When working with function values from a table, clearly indicate which values you're using and how you're using them
  • Always include units in your final answer when appropriate (especially for related rates and optimization problems)
  • Practice with past FRQs to get comfortable with the format and expectations
🔑 Key Formulas & Identities ▼
  • Squeeze Theorem: If \(f(x) \le h(x) \le g(x)\) and \(\lim_{x\to c} f(x) = L = \lim_{x\to c} g(x)\), then \(\lim_{x\to c} h(x) = L\)
  • Pythagorean Identities:
    • \(\sin^2 x + \cos^2 x = 1\)
    • \(1 + \tan^2 x = \sec^2 x\)
    • \(1 + \cot^2 x = \csc^2 x\)
  • Double Angle Formulas:
    • \(\sin 2x = 2\sin x \cos x\)
    • \(\cos 2x = \cos^2 x - \sin^2 x\)
    • \(\cos 2x = 2\cos^2 x - 1\)
    • \(\cos 2x = 1 - 2\sin^2 x\)
  • Common Maclaurin Series:
    • \(e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots\)
    • \(\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots\)
    • \(\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots\)
    • \(\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \dots \quad (-1 < x \le 1)\)
    • \(\frac{1}{1-x} = 1 + x + x^2 + x^3 + \dots \quad (|x| < 1)\)
  • Integration Substitution Techniques:
    • For integrals of the form \(\int f(ax+b) dx\), use \(u = ax+b\)
    • For integrals containing \(\sqrt{a^2-x^2}\), try \(u = a \sin \theta\) or \(u = a \cos \theta\)
    • For integrals containing \(\sqrt{a^2+x^2}\), try \(u = a \tan \theta\)
    • For integrals containing \(\sqrt{x^2-a^2}\), try \(u = a \sec \theta\)
Tags:
Shares: