📝 Natural exponential function e^x (16 MCQs)

📖 From Calculus • 1. Basics before calculus • 16 questions available

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What is Natural exponential function e^x?

Definition:
The natural exponential function $f(x) = e^x$ uses Euler's number $e \approx 2.71828$ , which is the unique base where the derivative equals the function itself, making it the most important exponential function in calculus and continuous growth.

Example:
$e^2 \approx 7.389$ , and its graph passes through (0,1), with slope 1 at $x=0$ , and increases faster than any polynomial.

Reason:
$e^x$ is central in natural processes like population growth, radioactive decay, and compound interest, due to its self-derivative property simplifying differential equations.

Easy

Medium

Hard

📝 All Natural exponential function e^x MCQs

Q1. What is the approximate value of the natural base e to six decimal places?

A.2.718281

B.2.718282 ✅

C.2.718283

D.2.718284

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The constant e is approximately 2.718282 when rounded to six decimal places, matching the commonly used value in calculations and aligning with the definition given in textbooks.

Q2. Which of the following limits equals e?

A.\

\\displaystyle \\lim_{n\\to\\infty}\\left(1+\\frac{1}{n}\\right)^{n}\

✅

B.\

\\displaystyle \\lim_{n\\to\\infty}\\left(1-\\frac{1}{n}\\right)^{n}\

C.\

\\displaystyle \\lim_{n\\to\\infty}\\left(1+\\frac{2}{n}\\right)^{n}\

D.\

\\displaystyle \\lim_{n\\to\\infty}n^{1/n}\

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The limit \ $\\lim_{n\\to\\infty}\\left(1+\\frac{1}{n}\\right)^{n}\$ is the classic definition of e, as the sequence converges to the natural base, whereas the other limits converge to different values.

Q3. If the slope of the tangent line to \ $y=b^{x}\$ at \ $x=0\$ equals the y‑coordinate at that point, which base b must be chosen?

A.1

B.e ✅

C.10

D.2

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The derivative of \ $b^{x}\$ is \ $b^{x}\\ln b\$ . At \ $x=0\$ the y‑coordinate is 1, so the slope equals \ $\\ln b\$ . Setting \ $\\ln b=1\$ gives \ $b=e\$ .

Q4. Given that the tangent line to \ $y=e^{x}\$ at \ $x=0\$ has slope 1, what is the slope of the tangent line to \ $y=e^{2x}\$ at \ $x=0\$ ?

A.1

B.2 ✅

C.e

D.0

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Differentiating \ $e^{2x}\$ yields \ $2e^{2x}\$ . Evaluating at \ $x=0\$ gives \ $2e^{0}=2\$ , so the slope is 2, larger than the slope of \ $e^{x}\$ at the same point.

Q5. Suppose \ $f(x)=e^{x}\$ and \ $g(x)=e^{-x}\$ . If the product \ $h(x)=f(x)g(x)\$ is differentiated, what is \ $h'(0)\$ ?

A.0 ✅

B.1

C.-1

D.2

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The product simplifies to \ $h(x)=e^{x}e^{-x}=1\$ . The derivative of a constant is 0, so \ $h'(0)=0\$ .

Q6. Assume a function \ $k(x)=b^{x}\$ has the property that the tangent line at any point \ $(a,b^{a})\$ passes through the origin. Which base b satisfies this condition?

A.e

B.1

C.\\sqrt{e}

D.None of the above ✅

💡 Difficulty: hard | ✅ Correct: D

📖 Explanation: Setting the tangent slope \ $b^{a}\\ln b\$ equal to the slope of the line through the origin \ $b^{a}/a\$ gives \ $\\ln b=1/a\$ . This cannot hold for all a, so no such base exists; thus none of the listed options work.

Q7. Which statement correctly compares the slopes at \ $x=0\$ of \ $y=e^{x}\$ and \ $y=2^{x}\$ ?

A.Both slopes are equal to 1

B.Slope of \

e^{x}\

is larger ✅

C.Slope of \

2^{x}\

is larger

D.Both slopes are zero

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: At \ $x=0\$ , the slope of \ $e^{x}\$ is \ $e^{0}=1\$ . For \ $2^{x}\$ the slope is \ $2^{0}\\ln2=\\ln2\\approx0.693\$ , which is smaller than 1, so the slope of \ $e^{x}\$ is larger.

Q8. The sequence \ $\\left(1+\\frac{1}{n}\\right)^{n}\$ approaches e as n\\to\\infty. Which of the following best describes its rate of convergence?

A.Linear convergence

B.Quadratic convergence

C.Exponential convergence

D.Sublinear (logarithmic) convergence ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: The convergence is slower than any polynomial rate; it improves like \ $1+\\frac{1}{2n}+\\dots\$ , which is sublinear (logarithmic) rather than linear or quadratic.

Q9. Find the base b such that the derivative of \ $b^{x}\$ at \ $x=0\$ equals 1.

A.e ✅

B.10

C.2

D.\\sqrt{e}

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The derivative at 0 is \ $\\ln b\$ . Setting \ $\\ln b=1\$ gives \ $b=e\$ , the unique base satisfying the condition.

Q10. Using the limit definition \ $e^{x}= \\lim_{n\\to\\infty}\\left(1+\\frac{x}{n}\\right)^{n}\$ , which step justifies that \ $\\exp(x+y)=\\exp(x)\\exp(y)\$ ?

A.Combine the limits into a single limit

B.Apply the binomial theorem

C.Use the property \

\\lim_{n\\to\\infty}(ab)^{n}= (\\lim a^{n})(\\lim b^{n})\

✅

D.Separate the exponent inside the limit

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Writing \ $\\exp(x)\\exp(y)=\\lim\\left(1+\\frac{x}{n}\\right)^{n}\\lim\\left(1+\\frac{y}{n}\\right)^{n}\$ and using \ $\\lim (ab)^{n}= (\\lim a^{n})(\\lim b^{n})\$ allows the product of limits to be combined, yielding \ $\\exp(x+y)\$ .

Q11. Why is the base e considered the ‘natural’ base for exponential functions?

A.Because it is the only base giving integer slopes

B.Because its derivative equals itself ✅

C.Because it is rational

D.Because it was discovered by Newton

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The defining property of the natural exponential is that its derivative is the same function, \ $\\frac{d}{dx}e^{x}=e^{x}\$ , making calculations and differential equations particularly simple.

Q12. Solve the differential equation \ $\\frac{dy}{dx}=y\$ with the initial condition \ $y(0)=1\$ .

A.y=e^{x} ✅

B.y=1

C.y=xe^{x}

D.y=\\ln x

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Separating variables gives \ $\\frac{dy}{y}=dx\$ . Integrating yields \ $\\ln y = x + C\$ . Using \ $y(0)=1\$ gives \ $C=0\$ , so \ $y=e^{x}\$ .

Q13. Continuous compounding at an annual rate r for t years yields amount \ $A = P e^{rt}\$ . If P=1000, r=5% and t=3, what is A (rounded to nearest dollar)?

A.1150

B.1162 ✅

C.1175

D.1200

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Compute \ $e^{0.05\\times3}=e^{0.15}\\approx1.161834\$ . Multiplying by 1000 gives \ $A\\approx1162\$ dollars, which is the nearest whole‑dollar amount.

Q14. For the function \ $f(x)=e^{x^{2}}\$ , what is \ $f'(1)\$ ?

A.e

B.2e ✅

C.e^{2}

D.2

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Using the chain rule, \ $f'(x)=e^{x^{2}}\\cdot 2x\$ . At \ $x=1\$ this becomes \ $2\\cdot e^{1}=2e\$ .

Q15. As \ $x\\to -\\infty\$ , what is the limit of \ $\\left(1+\\frac{1}{x}\\right)^{x}\$ ?

A.0

B.1

C.e

D.1/e ✅

💡 Difficulty: hard | ✅ Correct: D

📖 Explanation: For large negative \ $x\$ , the expression approaches \ $e^{-1}=1/e\$ because \ $\\left(1+\\frac{1}{x}\\right)^{x}=\\left(\\left(1+\\frac{1}{x}\\right)^{-x}\\right)^{-1}\$ and the inner term tends to \ $e\$ .

Q16. The natural logarithm function is defined as \ $\\ln x = \\log_{e} x\$ . Which identity correctly relates \ $\\ln\$ and the exponential function?

A.\

\\ln(e^{x}) = x\

✅

B.\

\\ln(e^{x}) = e^{x}\

C.\

\\ln(e^{x}) = x^{e}\

D.\

\\ln(e^{x}) = \\frac{1}{x}\

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: By definition, the natural log is the inverse of the natural exponential, so \ $\\ln(e^{x}) = x\$ for all real \ $x\$ .

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