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Applied Calculus Important Questions
English 14 leveled MCQs Free
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2 marks Short answer
Find the derivative of $f(x) = x^2$.
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$f'(x) = 2x$
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2 marks Short answer
Define the derivative of a function.
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The derivative of a function is the rate of change of the function with respect to its variable.
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2 marks Short answer
State the conditions for maxima and minima.
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The first derivative must be zero and the second derivative must be positive for minima and negative for maxima.
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2 marks Short answer
Find the derivative of $f(x) = x^4 + x^2$.
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$f'(x) = 4x^3 + 2x$
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2 marks Short answer
State Euler's theorem for homogeneous functions.
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Euler's theorem states that for a homogeneous function of degree $n$, $x_1 rac{\partial f}{\partial x_1} + x_2 rac{\partial f}{\partial x_2} + \ldots + x_k rac{\partial f}{\partial x_k} = n f(x_1, x_2, \ldots, x_k)$.
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2 marks Short answer
Define partial derivatives.
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Partial derivatives are the derivatives of a function with respect to one variable while keeping the other variables constant.
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2 marks Short answer
State the chain rule.
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The chain rule states that if $y = f(g(x))$, then $ rac{dy}{dx} = f'(g(x)) \cdot g'(x)$.
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2 marks Short answer
Define total derivatives.
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Total derivatives account for the change in a function with respect to all its variables simultaneously.
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2 marks Short answer
State the conditions for maxima and minima of functions of two variables.
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The first derivatives must be zero, and the second derivative test must indicate positive or negative definiteness.
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2 marks Short answer
State Lagrange's multiplier method.
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Lagrange's multiplier method is used to find the extrema of a function subject to constraints by introducing a multiplier for each constraint.
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2 marks Short answer
State the fundamental theorem of calculus.
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The fundamental theorem of calculus links the concept of differentiation and integration, stating that if $F$ is an antiderivative of $f$ on an interval $[a, b]$, then $\int_a^b f(x) \, dx = F(b) - F(a)$.
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2 marks Short answer
Define an indefinite integral.
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An indefinite integral is the integral of a function without specified limits, representing a family of functions plus a constant.
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2 marks Short answer
Define an improper integral.
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An improper integral is an integral where either the interval of integration is infinite or the integrand approaches infinity within the interval.
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2 marks Short answer
Write the reduction formula for $\int \sin^n x \, dx$.
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$\int \sin^n x \, dx = -\frac{1}{n} \sin^{n-1} x \cos x + \frac{n-1}{n} \int \sin^{n-2} x \, dx$.
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