Derivatives, Integrals, Limits & Continuity (Core Calculus Concepts)
The derivative of the function f(x) = 3x² - 5x + 2.
What is f'(x) = 6x - 5?
The rate the top of a 10-ft ladder slides down when the bottom is 6 ft from the wall and moving away at 1 ft/s.
What is -0.75 ft/s?
Evaluate the definite integral ∫₂⁵ (3x² - 4x + 1) dx using the Fundamental Theorem of Calculus.
What is 78?
Given the parametric equations x(t) = t² and y(t) = t³, find the derivative dy/dx as a function of t.
What is (3t²) / (2t) = (3t) / 2?
Find the radius of convergence for the power series ∑n=0∞x^n/n!.
What is infinite?
The integral of ∫ (2x³ - x + 4) dx with respect to x.
What is f(x)=(1/2)x⁴ - (1/2)x² + 4x + C?
The dimensions that maximize the area of a rectangular paddock with 500 ft of fencing.
What is 125 ft x 125 ft?
Approximate ∫₀³ x² dx using 3 equal subintervals and midpoints.
What is 8.75?
Evaluate the area enclosed by one loop of the rose curve r(θ) = 2sin(4θ).
What is π/4?
Write the first three non-zero terms of the Taylor series for e^x centered at x=0.
What is 1 + x + x²/2?
The limit as x approaches 3 of (x² - 9)/(x - 3).
What is 6?
The area under the curve f(x) = x² from x = 0 to x = 4.
What is 64/3 square units?
Given ∫₀⁴ f(x) dx = 5 and ∫₀⁴ g(x) dx = 3, find ∫₀⁴ [2f(x) - 3g(x)] dx.
What is 1?
Compute the divergence of the vector field F(x, y, z) = (xy, yz, zx).
What is x + y + z?
Use the ratio test to determine whether the series ∑n=1∞ 2^n/n² converges or diverges.
What is diverges?
Why the function f(x) = 1 / (x - 2) is not continuous at x = 2.
What is it is undefined at x = 2 due to a vertical asymptote?
How to find velocity and acceleration from a position function s(t).
What is velocity is s'(t) and acceleration is s''(t)?
Describe the relationship between indefinite integrals and antiderivatives.
What is an indefinite integral represents a general antiderivative plus a constant of integration?
Explain how vector calculus is used to analyze fluid flow.
What is it helps compute quantities like flow rate, circulation, and vorticity, describing how fluids move, speed up or slow down, and rotate in dynamic systems?
Explain how Taylor and Maclaurin series can be used to approximate functions.
What is they approximate functions using polynomials, which are easier to manipulate analytically?
The formal definition of a derivative using limits.
What is lim(h→0) [(f(a + h) - f(a))/h]?
The volume of the solid formed by revolving y = √x from x = 0 to x = 4 around the x-axis.
What is 8π cubic units?
Explain how the Fundamental Theorem of Calculus connects integration and differentiation.
What is it shows they are inverse processes: ∫ₐᵇ f(x) dx = F(b) - F(a) and d/dx ∫ₐˣ f(t) dt = f(x)? It connects the derivative and the integral by showing that integration can be undone by differentiation.
Discuss the advantages of using polar coordinates in certain integrations and graphing situations.
What is polar coordinates simplify calculations in cases with circular or spiral symmetry, making integrations more manageable and reducing complex double integrals into simpler single integrals?
Discuss the difference between absolute convergence and conditional convergence in series.
What is absolute convergence occurs if the series of absolute values converges, while conditional convergence does not?