Calculus practice problems with answers
Worksheets for every major calculus topic. Try each problem, then check the answer and the full step-by-step solution. When you want more, practice with new numbers every time.
Limits and continuity
Practice problems on evaluating limits, limits at infinity, L'Hôpital's rule and continuity.
Evaluating limits practice problems
\lim_{x\to a} f(x) = f(a) \text{ when } f \text{ is continuous at } a
12 limit practice problems with answers and step-by-step solutions: direct substitution, factoring 0/0, conjugates, trig limits and one-sided limits.
Limits at infinity practice problems
\lim_{x\to\infty}\frac{a x^n + \cdots}{b x^n + \cdots} = \frac{a}{b}
12 limits at infinity practice problems with answers and full solutions: rational functions, square roots, exponentials and horizontal asymptotes.
L'Hôpital's rule practice problems
\lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)}
12 L'Hôpital's rule practice problems with answers and full solutions, covering 0/0, ∞/∞, 0·∞, ∞−∞ and exponent forms like 1^∞ and 0^0.
Continuity practice problems
\lim_{x\to a} f(x) = f(a)
12 continuity practice problems with answers and full solutions: piecewise functions, finding k, types of discontinuity and the Intermediate Value Theorem.
Derivatives
Practice problems on the derivative rules, from the power rule to implicit differentiation.
Limit definition of the derivative practice problems
f'(x) = \lim_{h\to 0}\frac{f(x+h) - f(x)}{h}
12 limit definition of the derivative practice problems with answers and step-by-step solutions, from linear functions to 1/x, √x and tangent lines.
Power rule practice problems
\frac{d}{dx}\,x^n = n\,x^{n-1}
12 power rule practice problems with answers and step-by-step solutions: polynomials, negative and fractional exponents, roots and tangent lines.
Product and quotient rule practice problems
(fg)' = f'g + fg', \qquad \left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}
12 product rule and quotient rule practice problems with answers and full step-by-step solutions, including triple products and a table problem.
Derivatives of trig functions practice problems
\frac{d}{dx}\sin x = \cos x, \quad \frac{d}{dx}\cos x = -\sin x, \quad \frac{d}{dx}\tan x = \sec^2 x
12 derivatives of trig functions practice problems with answers and full solutions: sin, cos, tan, sec, csc, cot, with product, quotient and chain rules.
Chain rule practice problems
\frac{d}{dx}\,f(g(x)) = f'(g(x))\,g'(x)
12 chain rule practice problems with answers and full step-by-step solutions, from (3x+1)⁵ to nested functions like sin²(4x) and e^(cos x).
Derivatives of exponential and log functions practice problems
\frac{d}{dx}e^x = e^x, \qquad \frac{d}{dx}\ln x = \frac{1}{x}
12 practice problems on derivatives of exponential and logarithmic functions, with answers and full solutions: e^x, a^x, ln x, log rules and x^x.
Implicit differentiation practice problems
\frac{d}{dx}\big[f(y)\big] = f'(y)\,\frac{dy}{dx}
12 implicit differentiation practice problems with answers and full solutions: circles, products, trig and exponentials, tangent lines and y''.
Inverse trig derivatives practice problems
\frac{d}{dx}\arcsin x = \frac{1}{\sqrt{1 - x^2}}, \qquad \frac{d}{dx}\arctan x = \frac{1}{1 + x^2}
12 inverse trig derivative practice problems with answers and full solutions: arcsin, arccos, arctan and arcsec with the chain and product rules.
Applications of derivatives
Practice problems on related rates, optimization, curve sketching and linear approximation.
Related rates practice problems
x^2 + y^2 = z^2 \;\Rightarrow\; 2x\frac{dx}{dt} + 2y\frac{dy}{dt} = 2z\frac{dz}{dt}
10 related rates practice problems with answers and full solutions: balloons, ladders, cones, shadows, cars and angles, each with a clear setup.
Optimization practice problems
f'(x) = 0 \;\text{ or undefined} \;\Rightarrow\; \text{test critical points and endpoints}
10 calculus optimization practice problems with answers and full solutions: fences, boxes, cans, closest points, revenue and the fastest route.
Curve sketching practice problems
f'(x) > 0 \Rightarrow \text{increasing}, \qquad f''(x) > 0 \Rightarrow \text{concave up}
12 curve sketching practice problems with answers and full solutions: critical points, increasing and decreasing, extrema, concavity and asymptotes.
Linear approximation practice problems
L(x) = f(a) + f'(a)(x - a)
12 linear approximation practice problems with answers and full solutions: linearization, estimating roots and logs, differentials and error.
Mean Value Theorem practice problems
f'(c) = \frac{f(b) - f(a)}{b - a}
12 Mean Value Theorem practice problems with answers and full solutions: finding c, Rolle's theorem, checking hypotheses, bounds and counting roots.
Integrals
Practice problems on antiderivatives, Riemann sums, definite integrals and the Fundamental Theorem.
Antiderivative practice problems
\int x^n\,dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)
12 antiderivative practice problems with answers and step-by-step solutions: power rule, trig, exponentials and initial value problems.
Riemann sum practice problems
\int_a^b f(x)\,dx \approx \sum_{i=1}^{n} f(x_i^*)\,\Delta x, \quad \Delta x = \frac{b-a}{n}
10 Riemann sum practice problems with answers: left, right, midpoint and trapezoid sums, tables, over or under estimates and limits of sums.
Definite integral practice problems
\int_a^b f(x)\,dx = F(b) - F(a)
12 definite integral practice problems with answers and full solutions: polynomials, trig, symmetry, absolute value, geometry and integral properties.
Fundamental Theorem of Calculus practice problems
\frac{d}{dx}\int_a^{g(x)} f(t)\,dt = f(g(x))\,g'(x)
12 Fundamental Theorem of Calculus practice problems with answers: Part 1 derivatives with variable limits, Part 2 evaluation and accumulation functions.
Integration techniques
Practice problems on substitution, integration by parts, trig integrals and partial fractions.
U-substitution practice problems
\int f(g(x))\,g'(x)\,dx = \int f(u)\,du, \quad u = g(x)
12 u-substitution practice problems with answers and full step-by-step solutions, from (x²+1)⁵ to ln x / x, back-substitution and definite integrals.
Integration by parts practice problems
\int u\,dv = uv - \int v\,du
12 integration by parts practice problems with answers and full solutions: x eˣ, ln x, arctan x, repeated parts and the e^x sin x loop.
Trig integral practice problems
\sin^2 x = \frac{1 - \cos 2x}{2}, \quad \cos^2 x = \frac{1 + \cos 2x}{2}
12 trigonometric integral practice problems with answers: powers of sine and cosine, tangent and secant, half-angle and product-to-sum.
Trig substitution practice problems
\sqrt{a^2 - x^2}:\ x = a\sin\theta \quad \sqrt{a^2 + x^2}:\ x = a\tan\theta \quad \sqrt{x^2 - a^2}:\ x = a\sec\theta
12 trigonometric substitution practice problems with answers and full solutions for √(a²−x²), √(a²+x²), √(x²−a²) and completing the square.
Partial fractions practice problems
\frac{px + q}{(x - a)(x - b)} = \frac{A}{x - a} + \frac{B}{x - b}
12 partial fraction decomposition practice problems with answers: distinct and repeated factors, irreducible quadratics and long division first.
Applications of integrals
Practice problems on area, volume, improper integrals and differential equations.
Area between curves practice problems
A = \int_a^b \big(\text{top}(x) - \text{bottom}(x)\big)\,dx
10 area between curves practice problems with answers and full solutions: finding intersections, top minus bottom, crossing curves and integrating in y.
Volumes of revolution practice problems
V = \pi\int_a^b \big(R(x)^2 - r(x)^2\big)\,dx \qquad V = 2\pi\int_a^b x\,h(x)\,dx
11 volume of revolution practice problems with answers and full solutions using the disk, washer and shell methods, including other axes.
Improper integral practice problems
\int_a^{\infty} f(x)\,dx = \lim_{b\to\infty}\int_a^{b} f(x)\,dx
12 improper integral practice problems with answers: infinite limits, vertical asymptotes, convergence or divergence and the comparison test.
Differential equation practice problems
\frac{dy}{dx} = g(x)\,h(y) \;\Rightarrow\; \int\frac{dy}{h(y)} = \int g(x)\,dx
12 differential equation practice problems with answers: separable equations, exponential growth and decay, cooling, logistic growth and Euler's method.
Sequences and series
Practice problems on convergence tests, power series, Taylor series, parametric and polar curves.
Series convergence test practice problems
L = \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|: \quad L < 1 \text{ converges},\ L > 1 \text{ diverges}
12 series convergence practice problems with answers: nth term, p-series, geometric, comparison, integral, ratio, root and alternating series tests.
Power series practice problems
\frac{1}{1 - x} = \sum_{n=0}^{\infty}x^n, \quad |x| < 1
12 power series practice problems with answers: radius and interval of convergence, endpoint checks, and building series from 1/(1 − x).
Taylor series practice problems
f(x) = \sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x - a)^n
12 Taylor and Maclaurin series practice problems with answers: building series, Taylor polynomials, estimates with error bounds, limits and sums.
Parametric and polar practice problems
\frac{dy}{dx} = \frac{dy/dt}{dx/dt}, \qquad A = \frac{1}{2}\int_{\alpha}^{\beta} r^2\,d\theta
12 parametric and polar practice problems with answers: dy/dx, tangent lines, arc length, converting coordinates, polar slopes and polar area.
Multivariable calculus
Practice problems on partial derivatives, multiple integrals and optimization in several variables.
Partial derivative practice problems
f_x = \frac{\partial f}{\partial x}, \quad \nabla f = \langle f_x, f_y\rangle, \quad D_{\mathbf{u}}f = \nabla f\cdot\mathbf{u}
12 partial derivative practice problems with answers: first and second partials, gradients, directional derivatives, tangent planes and critical points.
Double integral practice problems
\iint_R f(x, y)\,dA = \int_a^b\!\int_{g_1(x)}^{g_2(x)} f(x, y)\,dy\,dx
12 double integral practice problems with answers: iterated integrals, general regions, switching the order, polar coordinates, volume and average value.
Lagrange multiplier practice problems
\nabla f = \lambda\,\nabla g, \qquad g(x, y) = k
12 Lagrange multiplier practice problems with answers: max and min on lines and circles, closest points, three variables and the meaning of lambda.
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