Find the derivative of f(x) = x³ ln(x).
Calculus
Question · Medium
Evaluate ∫₀¹ x·eˣ dx.
Chapter 1: Single-Variable Differential Calculus
1.1 Limits, Continuity, and Epsilon-Delta Definitions
Study guide
Let f(x) = (x² − 4)/(x − 2). What is lim_{x→2} f(x)?
Question · Easy
If f(x) = arctan(x), find f'(x).
Find the area enclosed between y = x² and y = x over [0, 1].
Which of the following is the general solution to the ODE dy/dx = 2xy?
1.2 Differentiation Rules, Implicit & Inverse Differentiation
For the series Σ_{n=1}^{∞} 1/n², which of the following is true?
Find ∂f/∂y for f(x, y) = x³y² + sin(xy).
Determine all critical points of f(x) = x³ − 3x + 1.
What is the radius of convergence of the power series Σ_{n=0}^{∞} nˣⁿ / n!?
Question · Hard
1.3 Mean Value Theorem, Rolle's Theorem, and Function Extrema
Use the second derivative test to classify the critical point x = 1 of f(x) = x⁴ − 4x + 2.
Evaluate the double integral ∫₀¹ ∫₀¹ (x + y) dy dx.
The Taylor series for eˣ centered at x = 0 is Σ_{n=0}^{∞} xⁿ/n!. What is the Taylor series for e^(−x²)?
Let A = [[1, 2], [3, 4]]. What is det(A)?
Algebra
1.4 L'Hôpital's Rule, Curve Sketching, and Optimization
Which of the following subsets of ℝ² is a subspace of ℝ²?
What is the order of the element 3 in the group (ℤ₇, +)?
How many prime numbers are there between 20 and 40?
The matrix A has eigenvalues 2 and 5. What is det(A)?
1.5 Functions, Graphs, Transformations, and Inverses
Find gcd(126, 84).
Let X be a Bernoulli random variable with P(X=1) = 0.3. What is Var(X)?
Additional Topics
Which of the following is NOT a valid metric on a nonempty set X?
How many elements are in the power set of a set with 4 elements?
1.6 Trigonometric Functions, Identities, and Equations
What is the modulus of the complex number z = 3 − 4i?
Which of the following sequences converges?
A graph G has 6 vertices and 9 edges. What is the sum of all vertex degrees?
Evaluate ∫ sec²(x) dx.
1.7 Coordinate Geometry, Lines, Circles, and Conics
Use L'Hôpital's rule to evaluate lim_{x→0} sin(x)/x.
Find the length of the curve y = (2/3)x^(3/2) from x = 0 to x = 3.
Which test correctly determines convergence of the series Σ_{n=1}^{∞} (−1)^n / n?
For f(x, y) = x²y − y³, find all critical points.
1.8 Parametric and Polar Curves
Solve the initial value problem: y' + 2y = 4, y(0) = 1.
Use the substitution u = x² to evaluate ∫ 2x·cos(x²) dx.
Find the gradient vector of f(x, y) = x²y + e^y at the point (1, 0).
The general solution to y'' − 5y' + 6y = 0 is:
Chapter 2: Single-Variable Integral Calculus
2.1 Riemann Sums and the Fundamental Theorem of Calculus
Which of the following statements about the series Σ_{n=1}^{∞} (−1)^(n+1)/√n is correct?
Evaluate ∫₀^π sin²(x) dx.
By Green's theorem, ∮_C (P dx + Q dy) = ∬_D (∂Q/∂x − ∂P/∂y) dA. For P = y, Q = x, and D the unit disk, the line integral equals:
If T: ℝ³ → ℝ² is a linear transformation with a 2×3 matrix representation, what is the maximum possible rank of T?
2.2 Advanced Integration Techniques
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