Engineering Mathematics Problems: Practice Questions

Here, you can find a list of engineering mathematics questions that can be treated as practice problems in Mid-Sem exam, Sep-Oct 2026.

Engineering Mathematics I Mid Term Practice Problems 2026

Q1: Find the following limits:

1. $\lim \limits_{x \to 0} [x] \quad$ [Solution]
2. $\lim \limits_{x \to 0} \dfrac{|x|}{x}$, if it exists. [Solution]
3. $\lim \limits_{x \to 0} \dfrac{\sqrt{1+x}-1}{x}\quad$ [Solution]
4. $\lim \limits_{x \to 0} \dfrac{1-\cos x}{x^2}$

Q2: Find the value of k for which the function f(x) = $\begin{cases} \dfrac{1-\cos 4x}{8x^2} \,\, \text{ if } x \neq 0 \\ k \quad \quad \quad \quad \text{ if } x=0 \end{cases}$ is continuous at x=0. [Solution]

Q3: Find the value of k for which the function f(x) = $\begin{cases} x^2+21 \,\, \text{ if } x < 3 \\ k \quad \quad \quad \text{ if } x=3 \\ x^3+3 \,\,\, \text{ if } x > 3 \end{cases}$ is continuous at x=3.

Q4: Find the derivative of the following functions at $x=0$, i.e, find $f'(0)$ (if possible)

1. f(x) = $\begin{cases} x \sin x \, \quad \text{ if } x \neq 0 \\ 0 \quad \quad \quad \,\, \text{ if } x=0 \end{cases}$. [Solution]
2. $f(x) = |x|$ [Solution]
3. $f(x) = x|x|$ [Solution]

Q5: Find the n^th derivatives for the functions [Solution]

1. $x^n$
2. $\dfrac{1}{x}$
3. $\ln x$
4. $\sin x$.

If $y=\sin (2x+\frac{\pi}{2})$. Find $y_3(\frac{\pi}{2})$.

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Q6: If y = tan^-1x, then show that (1+x²)y_n+1 +2nxy_n +n(n-1)y_n-1 = 0.

Q7: State Rolle’s theorem. Verify Rolle’s theorem for the functions below:

1. f(x) = x²+2 on [-2, 2] [Solution]
2. f(x) = sinx on [-π, π] [Solution]
3. f(x) = x³-x-1 on [0, 1] [Solution]

Q8: State Lagrange’s mean value theorem. Verify it for the functions

1. f(x) = x²+2x+3 on [4, 6] [Solution]
2. f(x) = x(x-1)(x-2) on [0, $\frac{1}{2}$] [Solution]

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Study Material

• nth Derivative
• Indeterminate Forms and L’Hospital Rule
• Rolle’s Theorem
• Taylor Series Expansion

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