Strengthen your understanding of every AP Calculus AB & BC topic — from limits and derivatives
to integrals, differential equations, series, and FRQ strategies.
Each quiz gives instant feedback, explanations, and real AP-style reasoning practice.
📘 Concept Mastery – Limits & Continuity (Advanced 10Q)
Master conceptual reasoning in limits, continuity, and one-sided behavior — as tested in the AP Calculus AB & BC exams.
1️⃣ Which of the following best defines \( \lim_{x \to c} f(x) \)?
A) The value that f(x) approaches as x → c
B) f(c)
C) That f(x) is continuous at c
D) The instantaneous rate of change
2️⃣ What does \( \lim_{x \to c^-} f(x) \) represent?
A) Limit as x approaches c from the left
B) Limit as x approaches c from the right
C) Average of left and right limits
D) f(c)
3️⃣ The limit \( \lim_{x \to 2} f(x) \) exists if and only if:
A) \( \lim_{x \to 2^-} f(x) = \lim_{x \to 2^+} f(x) \)
B) f(2) is defined
C) At least one one-sided limit exists
D) f is continuous at x=2
4️⃣ For what value of k is the function continuous at x=1?
\( f(x) = \begin{cases} x^2 + 2x – 3, & x < 1 \\ k, & x = 1 \\ 3x - 1, & x > 1 \end{cases} \)
\( f(x) = \begin{cases} x^2 + 2x – 3, & x < 1 \\ k, & x = 1 \\ 3x - 1, & x > 1 \end{cases} \)
A) k = 2
B) k = 3
C) k = 1
D) k = 4
5️⃣ If \( \lim_{x \to 3^-} f(x) = 1 \) and \( \lim_{x \to 3^+} f(x) = 5 \), what type of discontinuity occurs?
A) Jump discontinuity
B) Infinite discontinuity
C) Removable
D) Oscillating
6️⃣ Evaluate \( \lim_{x \to 0^+} \frac{1}{x} \).
A) \( +\infty \)
B) \( -\infty \)
C) 0
D) Does not exist
7️⃣ Compute \( \lim_{x \to 0} \frac{\sin(3x)}{x} \).
A) 3
B) 1/3
C) Does not exist
D) −3
8️⃣ The graph of f(x) shows a “hole” at x = 2 but the left and right sides meet. What kind of discontinuity is that?
A) Removable
B) Jump
C) Infinite
D) Oscillating
9️⃣ Evaluate \( \lim_{x \to 0} x^2 \sin\!\left(\frac{1}{x}\right) \).
A) 0
B) DNE
C) 1
D) ∞
🔟 If \( \lim_{x \to c} f(x) = L \) and \( \lim_{x \to c} g(x) = M \), which statement must be true?
A) \( \lim_{x \to c} [f(x)+g(x)] = L + M \)
B) \( L – M = 0 \)
C) \( f(c) = L \)
D) f and g are continuous
📗 Concept Mastery – Derivatives & Differentiability (Advanced 10Q)
Sharpen your conceptual understanding of derivatives, slopes, and differentiability — as tested in AP Calculus AB & BC exams.
1️⃣ The derivative \( f'(a) \) is defined as:
A) \( \lim_{h \to 0} \frac{f(a+h)-f(a)}{h} \)
B) \( \frac{f(b)-f(a)}{b-a} \)
C) \( \frac{dy}{dx} = \frac{\Delta y}{\Delta x} \)
D) \( f(b)-f(a) \)
2️⃣ If a function is differentiable at x = c, it must also be:
A) Continuous at x = c
B) Piecewise linear
C) Discontinuous at x = c
D) Having f′(c) = 0
3️⃣ The graph of f(x) has a sharp corner at x = 2. What conclusion is correct?
A) f is continuous but not differentiable at x = 2
B) f is discontinuous at x = 2
C) f′(2) exists and is 0
D) f has a removable discontinuity
4️⃣ If f and g are differentiable, the derivative of \( f(x)g(x) \) equals:
A) \( f'(x)g(x) + f(x)g'(x) \)
B) \( \frac{f′g − fg′}{g^2} \)
C) \( f′g \)
D) \( f′ + g′ \)
5️⃣ If \( y = \sin(3x) \), then \( \frac{dy}{dx} = ? \)
A) \( 3\cos(3x) \)
B) \( \cos(3x) \)
C) \( 3\sin(3x) \)
D) \( -3\cos(3x) \)
6️⃣ If f′(x) > 0 on an interval, which of the following is true?
A) f is increasing
B) f is decreasing
C) f is concave up
D) f is constant
7️⃣ For \( f(x) = |x| \), what happens at x = 0?
A) Continuous but not differentiable
B) Discontinuous
C) Differentiable
D) Concave down
8️⃣ If f and g are inverses and f(2) = 5, f′(2) = 4, then g′(5) = ?
A) 1/4
B) 4
C) −4
D) −1/4
9️⃣ Suppose \( f(x) = x^3 + 2x^2 \). At which x-value is the tangent line horizontal?
A) x = 0 and x = −4/3
B) x = 1
C) x = −2
D) x = 2
🔟 Which of the following statements is always true for differentiable functions?
A) Every differentiable function is continuous.
B) Every continuous function is differentiable.
C) Every differentiable function is even.
D) Every differentiable function is periodic.
📘 Concept Mastery – Applications of Derivatives (Advanced 10Q)
Master how derivatives describe motion, optimization, and curvature — exactly as tested in AP Calculus AB & BC exams.
1️⃣ If f′(x) > 0 for all x in (a,b), which statement is true?
A) f is increasing on (a,b)
B) f is decreasing on (a,b)
C) f is concave up
2️⃣ Mean Value Theorem (MVT) guarantees that:
A) There exists c such that f′(c) equals the average rate of change
B) f′(c)=0 always
C) Only continuity is required
D) Every point satisfies f′(x)=(f(b)−f(a))/(b−a)
3️⃣ A function has f′(x)=0 only at x=2, and f″(x)=−3. What can be concluded?
A) Local maximum at x=2
B) Local minimum at x=2
C) Point of inflection
D) Function is decreasing at x=2
4️⃣ If f′ changes from positive to negative at x=4, then:
A) f has a local maximum at x=4
B) f has a local minimum
C) Point of inflection
D) No extremum
5️⃣ If f″(x)>0 on an interval, then:
A) f is concave up
B) f is concave down
D) f is increasing
6️⃣ A point of inflection occurs when:
A) f″ changes sign
B) f′=0
C) f is continuous
D) f″=0 only
7️⃣ A ball’s position is s(t). If s′(t)=0 and s″(t)>0, what happens?
B) Ball at highest point
C) Moving fastest
D) Constant velocity
8️⃣ Optimization: For a rectangle with fixed perimeter P, the area is maximized when:
A) It is a square
B) It is a long rectangle
C) It is circular
D) One side = twice the other
9️⃣ Related Rates: If a circle’s radius increases at 2 cm/s, how fast does its area increase when r=5?
A) 20π cm²/s
B) 8π cm²/s
C) 10π cm²/s
D) 5π cm²/s
🔟 If f′(x)=0 and f″(x)=0 at x=c, what can be concluded?
A) Test higher derivatives; inconclusive
B) Always inflection
C) Always minimum
D) Always maximum
📙 Concept Mastery – Integrals & The Fundamental Theorem (Advanced 10Q)
Deepen your understanding of accumulation, area, and the relationship between derivatives and integrals — essential for the AP Calculus AB & BC exams.
1️⃣ The definite integral \( \int_a^b f(x)\,dx \) represents:
A) Net signed area under f(x) between x=a and x=b
B) Total area only
C) Family of antiderivatives
D) Average value of f(x)
2️⃣ If \( f(x) \ge 0 \) for x in [a,b], then \( \int_a^b f(x)\,dx \) equals:
A) The geometric area under f(x)
B) Net area (may be negative)
C) 0
D) Mean value of f
3️⃣ The Fundamental Theorem of Calculus (Part I) states:
A) The derivative of an accumulation function equals the original function.
B) The integral of f′(x) equals f(b)−f(a)
C) The average of f equals its rate of change
D) f(x) must be continuous and differentiable everywhere
4️⃣ FTC Part II allows us to evaluate \( \int_a^b f(x)\,dx \) by:
A) Finding any antiderivative F and computing F(b)−F(a)
B) Approximating with rectangles
C) Differentiating f(x)
D) Using u-substitution
5️⃣ If \( F(x)=\int_1^x t^2 dt \), what is F′(x)?
A) \( x^2 \)
B) \( 2x \)
C) 0
D) \( \frac{x^3}{3} \)
6️⃣ The expression \( \int_a^b |f(x)| dx \) represents:
A) Total geometric area
B) Net signed area
C) Average value
D) Antiderivative family
7️⃣ Which statement about substitution is true?
A) It is the reverse of the Chain Rule
B) It’s the reverse of the Product Rule
C) Reverse of the Quotient Rule
D) Reverse of L’Hôpital’s Rule
8️⃣ The average value of a continuous function on [a,b] is:
A) \( \frac{1}{b-a}\int_a^b f(x)dx \)
B) \( \int_a^b f(x)dx \)
C) \( f′(x) = f(b)-f(a) \)
D) \( f′(c) = \frac{f(b)-f(a)}{b-a} \)
9️⃣ If the velocity of a particle is negative, then the integral of velocity over an interval gives:
A) Negative displacement
B) Total distance
C) Average speed
D) Time elapsed
🔟 A right Riemann sum with n rectangles approximates:
A) The definite integral using right-endpoint values
B) Left endpoint approximation
C) Midpoint rule
D) Trapezoidal rule
📒 Concept Mastery – Differential Equations & Modeling (Advanced 10Q)
Understand how differential equations describe change, predict behavior, and connect slope fields to real-world models — a crucial skill in AP Calculus AB/BC.
1️⃣ A differential equation of the form \( \frac{dy}{dx} = ky \) represents:
A) Exponential growth or decay
B) Logistic growth
C) Linear change
D) Inverse proportional change
2️⃣ The general solution to \( \frac{dy}{dx} = ky \) is:
A) \( y = Ce^{kx} \)
B) \( y = \frac{C}{1+e^{-kx}} \)
C) \( y = kx + C \)
D) \( y = Cx^k \)
3️⃣ In a slope field, horizontal segments (zero slopes) indicate:
A) Equilibrium points (dy/dx=0)
B) Growth regions
C) Decreasing intervals
D) Undefined region
4️⃣ A solution curve passing through a region where slopes are negative will:
A) Decrease
B) Increase
C) Concave up
D) Stay constant
5️⃣ The logistic differential equation has the form:
A) \( \frac{dy}{dt} = ky(1 – \frac{y}{L}) \)
B) \( \frac{dy}{dt} = ky \)
C) \( \frac{dy}{dt} = \frac{k}{y} \)
D) \( \frac{dy}{dt} = ky^2 \)
6️⃣ In the logistic model, as time → ∞, y approaches:
A) The carrying capacity L
B) Infinity
C) 0
D) dy/dt = 0
7️⃣ Solving \( \frac{dy}{dx} = 2x \) by separation gives:
A) \( y = x^2 + C \)
B) \( y = 2x + C \)
C) \( y = \frac{x^2}{2} + C \)
D) \( y = 2x^2 \)
8️⃣ A direction field helps visualize:
A) Behavior of solutions without solving explicitly
B) The actual graph of f(x)
C) Numerical values of y
D) Limits at infinity
9️⃣ In population models, when population is small compared to carrying capacity, logistic growth behaves like:
A) Exponential growth
B) Linear growth
C) Exponential decay
D) Random pattern
🔟 In solving separable DEs, after separating variables, the next step is to:
A) Integrate both sides
B) Differentiate both sides
C) Multiply both sides by dx/dy
D) Substitute y=x
📘 Concept Mastery – BC Exclusive: Parametric, Polar & Series (Advanced 10Q)
Challenge your understanding of BC-only concepts — from motion in parametric form and polar curves to convergence and power series mastery.
1️⃣ For a parametric curve \( x = t^2 \), \( y = t^3 \), \( \frac{dy}{dx} \) equals:
A) \( \frac{3t}{2} \)
B) \( \frac{3x}{2} \)
C) \( \frac{2t}{3} \)
D) \( 3t^2 \)
2️⃣ For the same curve, the second derivative \( \frac{d^2y}{dx^2} \) equals:
A) \( \frac{3}{4t} \)
B) \( \frac{3t}{2} \)
C) \( \frac{9t^2}{2} \)
D) \( \frac{4t}{3} \)
3️⃣ The slope of a polar curve \( r = 2\sin\theta \) is given by:
A) \( \frac{r′\sin\theta + r\cos\theta}{r′\cos\theta – r\sin\theta} \)
B) \( \frac{dy}{d\theta} / \frac{dx}{d\theta} \)
C) \( \frac{\sin\theta}{\cos\theta} \)
D) \( \frac{r′\cos\theta – r\sin\theta}{r′\sin\theta + r\cos\theta} \)
4️⃣ The area enclosed by a polar curve \( r = f(\theta) \) from \( a \) to \( b \) is:
A) \( \frac{1}{2}\int_a^b [f(\theta)]^2 d\theta \)
B) \( \int_a^b [f(\theta)]^2 d\theta \)
C) \( \int_a^b \sqrt{(r′)^2 + r^2} d\theta \)
D) \( \int r d\theta \)
5️⃣ For a particle moving along a parametric curve \( x(t), y(t) \), the speed is:
A) \( \sqrt{(dx/dt)^2 + (dy/dt)^2} \)
B) \( \sqrt{(d^2x/dt^2)^2 + (d^2y/dt^2)^2} \)
C) \( dx/dt + dy/dt \)
D) \( |dx/dt – dy/dt| \)
6️⃣ The geometric series \( \sum_{n=0}^{\infty} ar^n \) converges when:
A) \( |r| < 1 \)
B) \( |r| > 1 \)
C) \( r=1 \)
D) \( r=-1 \)
7️⃣ The nth-term test says a series \( \sum a_n \) diverges if:
A) \( \lim_{n\to\infty} a_n \neq 0 \)
B) \( \lim a_n = 0 \) ensures convergence
C) \( a_n > 0 \)
D) \( \lim a_n = \infty \) ensures convergence
8️⃣ Alternating Series Test requires:
A) Alternating signs, decreasing magnitude, and limit → 0
B) Alternating signs only
C) Only decreasing sequence
D) Limit = 1
9️⃣ The Ratio Test: A series \( \sum a_n \) converges absolutely if:
A) \( \lim |a_{n+1}/a_n| < 1 \)
C) \( \lim |a_{n+1}/a_n| = 1 \)
D) \( \lim a_{n+1}/a_n = 0 \)
🔟 The Taylor series for \( e^x \) centered at 0 is:
A) \( \sum_{n=0}^{\infty} \frac{x^n}{n!} \)
B) \( \sum x^n \)
C) \( \sum r^n \)
D) \( \sum (x−1)^n \)
📗 Concept Mastery – FRQ Strategy: Mixed Reasoning (Advanced 10Q)
Practice conceptual FRQ-style thinking — where explanation, justification, and interpretation matter more than computation.
1️⃣ A particle moves along a line with velocity \( v(t) = t^2 – 4t + 3 \). The particle changes direction when:
A) \( t = 1 \) and \( t = 3 \)
B) \( t = 2 \)
C) When a(t)=0
D) Never
2️⃣ Given that \( f'(x) > 0 \) and \( f”(x) < 0 \), the function f(x) is:
A) Increasing and concave down
C) Increasing and concave up
D) Decreasing and concave up
3️⃣ If \( f'(x) = 0 \) at x=c but changes from negative to positive, then f(x) has:
A) A local minimum
B) A local maximum
C) No extremum
D) Point of inflection
4️⃣ If \( f'(x) = 0 \) at x=c and \( f”(c) > 0 \), then:
A) f has a local minimum
B) f has a local maximum
C) Point of inflection
D) No conclusion possible
5️⃣ If \( f(x) \) is continuous on [a,b], then by the Mean Value Theorem there exists c in (a,b) such that:
A) \( f′(c) = \frac{f(b)−f(a)}{b−a} \)
B) \( f(c) = \frac{f(a)+f(b)}{2} \)
C) \( f′(x) = f(x) \)
D) \( f(b)−f(a) = \int_a^b f′(x)dx \)
6️⃣ If g(x) = ∫₀ˣ f(t) dt, which of the following is true by the Fundamental Theorem?
A) \( g′(x) = f(x) \)
B) \( g′(x) = ∫₀ˣ f′(t)dt \)
C) \( g(x) = f′(x) \)
D) \( g′(x) = 0 \)
7️⃣ For the curve \( y = x^3 – 3x + 1 \), the inflection point occurs when:
A) \( x = 0 \)
B) \( x = ±1 \)
C) \( x = 2 \)
D) \( x = 3 \)
8️⃣ For a function f(x), concavity changes when:
A) \( f”(x) \) changes sign
B) \( f′(x) \) changes sign
C) f(x) continuous
D) \( f(x) = 0 \)
9️⃣ In an FRQ, to justify a local maximum using first derivative language, you must write:
A) “Since f′(x) changes from + to − at x=c, f has a local max.”
B) “Since f′ changes from − to +, f has a local max.”
C) “f′(x)=0 at x=c.”
D) “f′(x)=0, so local max.”
🔟 In a motion problem, if v(t)>0 and a(t)<0, the particle is:
A) Moving right and slowing down
B) Moving right and speeding up
D) At rest
Want to challenge yourself further? Try the Mock Exam or Calculator Skills section next.