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

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AP® Calculus AB Score Calculator

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Section I: Multiple-Choice 0/45
Section II: Free Response Questions
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1
Keep working on those derivatives and integrals!
MCQ Score (scaled to 54)
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Total Composite Score
0/108
020406080108
Score Thresholds (approximate):
1 (0-36) • 2 (37-44) • 3 (45-56) • 4 (57-68) • 5 (69+)
Official AP Calculus AB Practice

AP Calculus AB – 2025 Cheat Sheet

📊 Unit 1: Limits & Continuity ▼
  • limx→c f(x) is the value f(x) approaches when x → c from BOTH sides.
  • limx→c+ f(x) or limx→c- f(x) is the value f(x) approaches when x → c from ONLY the right (if +) or left (if -) side.
  • Limit Properties:
    • limx→c (af(x)) = a limx→c f(x)
    • limx→c (f(x) ± g(x)) = limx→c f(x) ± limx→c g(x)
    • limx→c (f(x)g(x)) = (limx→c f(x))(limx→c g(x))
    • limx→c f(x)/g(x) = (limx→c f(x)) / (limx→c g(x)), provided limx→c g(x) ≠ 0
    • limx→c f(g(x)) = f(limx→c g(x)) (if f is continuous at limx→c g(x))
  • Methods to algebraically simplify limits if you can't directly plug in: Completing the square, Rationalization, Factoring.
  • Order of growth rates from fastest to slowest: xx, x!, ax, xn, xln(x), ln(x).
  • For f(x)/g(x), if highest power of f > highest power of g: infinite limit DNE; if <, HA at y=0; if =, HA at y = ratio of first terms.
  • Continuity if f(c) = limx→c f(x).
  • Types of Discontinuities:
    • Removable: hole (factor cancels)
    • Non-removable: Asymptote (denominator is zero after simplifying), Jump (piecewise where y-values different from left and right).
  • IVT PROBLEMS: Write "Since f(x) is continuous on [a,b] and k is between f(a) and f(b), by the IVT there is a c in (a,b) such that f(c) = k".
📈 Unit 2: Fundamentals of Differentiation ▼
  • Average Rate of Change (AROC): AROC = (f(x+h) - f(x))/h or (f(b) - f(a))/(b-a)
  • Definition of Derivative: f'(x) = limh→0 (f(x+h) - f(x))/h = limz→x (f(z) - f(x))/(z-x)
  • When estimating f'(c) from a table, straddle both sides and use AROC. From a graph, f'(c) is the slope of the tangent line at x=c.
  • All differentiable functions are continuous, but not all continuous functions are differentiable (e.g., corners, cusps, vertical tangents).
  • Power Rule: d/dx (xn) = nxn-1
  • d/dx (c) = 0 (c is a constant)
  • d/dx (f(x) ± g(x)) = f'(x) ± g'(x)
  • d/dx (kf(x)) = kf'(x)
  • Trig Derivatives:
    • d/dx (sin(x)) = cos(x)
    • d/dx (cos(x)) = -sin(x)
    • d/dx (tan(x)) = sec2(x)
    • d/dx (cot(x)) = -csc2(x)
    • d/dx (sec(x)) = sec(x)tan(x)
    • d/dx (csc(x)) = -csc(x)cot(x)
  • d/dx (ex) = ex
  • d/dx (ln(x)) = 1/x
  • Product Rule: d/dx (f(x)g(x)) = f'(x)g(x) + f(x)g'(x)
  • Quotient Rule: d/dx (f(x)/g(x)) = (f'(x)g(x) - f(x)g'(x)) / (g(x))2
🔄 Unit 3: Composite, Implicit, & Inverse Functions ▼
  • d/dx (ax) = axln(a)
  • d/dx (loga(x)) = 1/(x ln(a))
  • Chain Rule: d/dx f(g(x)) = f'(g(x)) · g'(x). (Derivative of the outside, evaluated at the inside, times derivative of the inside).
  • Implicit Differentiation: Differentiate each term with respect to the independent variable (usually x). When differentiating a term with y, multiply by dy/dx. Then solve for dy/dx.
    Ex: d/dx (xy) = 1·y + x·(dy/dx) = y + x(dy/dx).
  • Inverse Function Derivatives:
    • d/dx (arcsin(x)) = 1/√(1-x2)
    • d/dx (arccos(x)) = -1/√(1-x2)
    • d/dx (arctan(x)) = 1/(1+x2)
    • d/dx (arccot(x)) = -1/(1+x2)
    • d/dx (arcsec(x)) = 1/(|x|√(x2-1))
    • d/dx (arccsc(x)) = -1/(|x|√(x2-1))
    (Derivatives of inverse trig cofunctions are the negative of the derivative of the other 3 inverse trig functions).
  • Higher order derivatives: Just repeat the differentiation process.
  • Second derivatives of implicit functions are functions of x, y, dy/dx. Substitute the first derivative expression for dy/dx.
⚙️ Unit 4: Contextual Applications of Differentiation ▼
  • The derivative of a function is the rate of change of that function.
  • If you are being asked about the rate of change of a rate of change, that's basically the derivative of f'(x), or f''(x).
  • Particle motion: d2x/dt2 = dv/dt = a(t). (Position x(t), velocity v(t), acceleration a(t)).
  • Steps for Related Rates:
    1. Draw picture.
    2. List knowns and unknowns (variables and their rates).
    3. Write an equation to model the situation (DO NOT PLUG IN VALUES THAT CHANGE YET).
    4. Differentiate with respect to time (d/dt).
    5. Substitute known values for variables and their rates.
    6. Solve for desired rate.
  • Linearization (Tangent Line Approximation): L(x) = f(c) + f'(c)(x-c). So f(x) ≈ L(x) for x near c.
    Or f(c+a) ≈ f(c) + f'(c)a.
  • L'Hopital's Rule (LHR): If limx→c f(x)/g(x) is of indeterminate form 0/0 or ∞/∞, then limx→c f(x)/g(x) = limx→c f'(x)/g'(x), provided the latter limit exists.
    • Sometimes you may need to use LHR multiple times.
    • Remember to plug in for the limit to check for indeterminate form before doing LHR!
📉 Unit 5: Analytical Applications of Differentiation ▼
  • MVT PROBLEMS: Write "Since f(x) is continuous on [a,b] and differentiable on (a,b), there exists a c in (a,b) where f'(c) = (f(b) - f(a))/(b-a) by the MVT."
  • Rolle's Theorem is MVT where f(b) = f(a), so f'(c)=0.
  • EVT (Extreme Value Theorem) PROBLEMS: Write "Since f(x) is continuous on [a,b], by the EVT, there exists at least one global maximum and one global minimum on [a,b]." (Check critical points and endpoints).
  • Critical Points: Where f'(x) = 0 or undefined.
  • Local Extrema: Point that is greater/less than surrounding points, always at critical points or endpoints (if on a closed interval).
    • First Derivative Test: If f'(x) changes from + to - at c, local max. If - to +, local min.
  • Global Extrema: Greatest or Least value of function on an interval. Found at critical points or endpoints.
  • f'(x) > 0: f is increasing; f'(x) < 0: f is decreasing.
  • If asking for whether the rate of change of f(x) is increasing or decreasing, this is asking for the sign of the second derivative (f''(x)).
  • Where f'(x) = 0, if f''(x) > 0: local min; if f''(x) < 0: local max (Second Derivative Test). If f''(x) = 0, test is inconclusive.
  • ENDPOINTS CAN BE EXTREMA TOO, REMEMBER THEM WHEN FINDING GLOBAL EXTREMA.
  • f''(x) > 0: f is concave up (ccu); f''(x) < 0: f is concave down (ccd).
  • Inflection point: where concavity changes (f''(x) = 0 or undefined, AND f''(x) changes sign).
  • Steps for Optimization:
    1. Draw picture (if applicable).
    2. Write primary equation (quantity to be optimized).
    3. Write constraint equation (if applicable), solve for one variable, and plug into primary equation.
    4. Find extrema of primary equation (using critical points and first/second derivative test) and solve for variables. Answer the question asked.
∫ Unit 6: Integration of Accumulation of Change ▼
  • Accumulation/Integral = area between a rate of change graph and the x-axis.
  • If area is below x-axis, then it's accumulating negative area.
  • Can use geometry to find integral from a graph (look for circles, triangles, squares, and places where positive and negative area cancel out).
  • When a function is split into multiple subdivisions Δx is the interval width.
    • LRS (Left Riemann Sum): ∑ f(xi)Δx (use left endpoint of each subinterval)
    • RRS (Right Riemann Sum): ∑ f(xi+1)Δx (use right endpoint)
    • MRS (Midpoint Riemann Sum): ∑ f((xi+xi+1)/2)Δx (use midpoint)
    • Trapezoidal Rule: ∑ 1/2 (f(xi) + f(xi+1))Δx
  • Riemann Sum (Formal Definition): ∫ab f(x)dx = limn→∞ ∑k=1n f(xk*)Δx where Δx = (b-a)/n and xk* is a point in the kth subinterval.
  • FTOC Pt 1 (Definite Integrals): If F'(x) = f(x), then ∫ab f(x)dx = F(b) - F(a).
  • FTOC Pt 2: d/dx ∫a(x)b(x) f(t)dt = f(b(x))·b'(x) - f(a(x))·a'(x).
  • Integrals can be used if there are jump or removable discontinuities (but not vertical asymptotes within the interval of integration).
  • Properties of Integrals:
    • ∫ab (f(x) ± g(x))dx = ∫ab f(x)dx ± ∫ab g(x)dx
    • ∫ab cf(x)dx = c∫ab f(x)dx
    • ∫aa f(x)dx = 0
    • ∫ab f(x)dx = - ∫ba f(x)dx
    • If b is between a and c: ∫ac f(x)dx = ∫ab f(x)dx + ∫bc f(x)dx
  • U-substitutions are your friend, use them!
    • Good substitutions for u: inner functions, functions with higher powers, denominator/numerator, an antiderivative present in the function to integrate.
    • May need long division or completing the square before u-sub.
    • Change limits of integration if definite integral, or sub back u if indefinite.
🔣 Unit 7: Differential Equations ▼
  • Slope Fields show tangent line segments (slopes) to the particular solution through that point. Calculated using the given dy/dx.
  • If you can write the differential equation in the form g(y)dy = f(x)dx, it's a separation of variables problem. Integrate both sides, then solve for y if possible (or leave implicit). Don't forget "+ C".
  • When all constants from antidifferentiation are replaced with appropriate values (using an initial condition), you get a particular solution.
  • Exponential Growth/Decay: dy/dt = ky → y = Cekt where C = y(0) (initial value).
📐 Unit 8: Applications of Integration ▼
  • Average Value of f(x) on [a,b]: (1/(b-a)) ∫ab f(x)dx.
  • AVT (Average Value Theorem / MVT for Integrals) PROBLEMS: Write "Since f(x) is continuous on [a,b], by the AVT, there must be a c in (a,b) where f(c) = (1/(b-a)) ∫ab f(x)dx."
  • If v(t) is velocity, then ∫ab v(t)dt = x(b) - x(a) (displacement).
  • DISPLACEMENT ≠ DISTANCE TRAVELED, just like how VELOCITY ≠ SPEED.
  • Speed = |v(t)|.
  • Distance traveled = ∫ab |v(t)|dt.
  • Area between curves:
    • If integrating with respect to x (dx): ∫ab (Top function - Bottom function)dx.
    • If integrating with respect to y (dy): ∫cd (Right function - Left function)dy.
    • Find intersection points to determine limits of integration and where functions switch top/bottom or right/left.
  • Volume by Known Cross-Sections:
    • Integrand is A(x) or A(y), the area of one cross-section.
    • If cross-sections are perpendicular to x-axis: ∫ab A(x)dx.
    • If cross-sections are perpendicular to y-axis: ∫cd A(y)dy.
  • Volume by Revolution (Disk/Washer Method):
    • Disk Method (solid, no hole): π ∫ R2 dx (or dy). R is radius from axis of revolution to outer curve.
    • Washer Method (hole): π ∫ (Router2 - Rinner2) dx (or dy). Router is radius from axis to outer curve, Rinner is radius from axis to inner curve.
    • Choose dx or dy based on whether the axis of revolution is horizontal (dy) or vertical (dx) if rectangles are drawn perpendicular to axis, or vice-versa for parallel (shell method - not typically AB). Usually perpendicular.
📝 FRQ Tips ▼
  • Work on the parts you know you can do first before moving onto other parts!
  • Be sure to show all your work still, even though it is a shorter test.
  • Shorthand like IVT, MVT, FTOC for Intermediate Value Theorem, Mean Value Theorem, Fundamental Theorem of Calculus is fine!
  • Don't simplify your answers if you don't need to! You don't want to unnecessarily lose points on steps you don't need to do!
  • Always state theorems properly when using them. For example, when using the Mean Value Theorem, first verify the conditions (continuity on [a,b] and differentiability on (a,b)).
  • When asked to "find" something, always show your work. When asked to "explain" something, write in complete sentences.
  • Pay attention to the units of measurement and include them in your final answer.
  • When working with related rates or optimization problems, draw a picture and label it clearly.
  • If asked for an approximation using Riemann sums, clearly identify which method you're using (left, right, midpoint, or trapezoidal).
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