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MathCalculus

Rolle's Theorem: Verify f(x)=2x²−8x+6 on [1,3] and Find c

Verify Rolle's theorem for 2x²−8x+6 on [1,3] and find the only interior point with derivative zero.

MathCalculus

Limit Laws: Evaluate lim(x→1/2) (2x²+3x−2)/(2x−1)

Identify the 0/0 form, factor the numerator, and evaluate the rational-function limit as x approaches 1/2.

MathCalculus

Derivatives of Trigonometric Functions: Differentiate sin x cos x

Differentiate sin x cos x with the product rule and check the result by a double-angle identity.

MathCalculus

Derivative for Exponential Products: Differentiate x e^(2x)

Find the derivative of x e^(2x) using product and chain rules.

MathCalculus

Inverse Derivative of 9 − x² on 0 ≤ x ≤ 3 at a = 2

Find f prime at 2, the inverse formula on the stated domain, and its derivative at y=5.

MathCalculus

Related Rates Calculus: Current Change as Voltage Falls and Resistance Rises

Find the instantaneous current rate when voltage falls and resistance rises.

MathCalculus

Trig Substitution: Evaluate ∫√(4 − x²) dx

Use x = 2 sin θ to evaluate OpenStax Calculus Volume 2 exercise 136, then return the answer to x and check its domain.

MathCalculus

Partial Fractions: Integrate 1/[(x − 3)(x − 2)]

Decompose the rational integrand in OpenStax Calculus Volume 2 exercise 196, integrate the terms, and keep the excluded poles explicit.

MathCalculus

Inverse Trig Integrals: Find ∫dx/(25 + 4x²)

Use a scaling substitution to evaluate OpenStax Checkpoint 1.42 and confirm the inverse tangent coefficient by differentiation.