FE Other Disciplines (NCEES Fundamentals of Engineering) — All Questions

What is the derivative of f(x) = 3x^3 - 5x evaluated at x = 2?

• a.31✓
• b.27
• c.36
• d.19

Differentiate term by term: f'(x) = 9x^2 - 5. At x = 2, f'(2) = 9(4) - 5 = 36 - 5 = 31. The power rule d/dx(x^n) = n·x^(n-1) is the core single-variable calculus tool tested on the FE.

What is the determinant of the 2x2 matrix [[4, 3], [2, 5]]?

• a.26
• b.7
• c.20
• d.14✓

For a 2x2 matrix [[a, b], [c, d]], the determinant is ad - bc. Here (4)(5) - (3)(2) = 20 - 6 = 14. Determinants indicate whether a linear system has a unique solution (nonzero determinant).

Evaluate the definite integral of sin(x) from 0 to pi.

• a.1
• b.2✓
• c.pi
• d.0

The antiderivative of sin(x) is -cos(x). Evaluating from 0 to pi gives [-cos(pi)] - [-cos(0)] = -(-1) - (-(1)) = 1 + 1 = 2. This is a standard application of the Fundamental Theorem of Calculus.

What is the derivative of f(x) = ln(x) evaluated at x = 2?

• a.0.693
• b.0.5✓
• c.-0.25
• d.2

The derivative of ln(x) is 1/x, so f'(2) = 1/2 = 0.5. The value 0.693 is ln(2) itself, not its derivative.

What is the derivative of f(x) = e^(2x) evaluated at x = 0?

• a.4
• b.0
• c.1
• d.2✓

By the chain rule d/dx[e^(2x)] = 2e^(2x). At x = 0 this is 2·e^0 = 2·1 = 2. The factor of 2 comes from differentiating the inner function 2x.

Evaluate the definite integral of x^2 from 0 to 2.

• a.8
• b.4
• c.1.333
• d.2.667✓

The antiderivative of x^2 is x^3/3. Evaluating from 0 to 2 gives 2^3/3 - 0 = 8/3 = 2.667. This applies the power rule for integration and the Fundamental Theorem of Calculus.

Evaluate the definite integral of 1/x from 1 to e.

• a.1.718
• b.2.718
• c.0.368
• d.1✓

The antiderivative of 1/x is ln(x). Evaluating from 1 to e gives ln(e) - ln(1) = 1 - 0 = 1. This is a standard natural-log integral.

What is the derivative of f(x) = cos(x) evaluated at x = 0?

• a.0✓
• b.0.5
• c.-1
• d.1

The derivative of cos(x) is -sin(x). At x = 0 this is -sin(0) = 0. Note cos(0) = 1, but the derivative evaluates the sine, not the cosine.

What is the derivative of f(x) = x^2·e^x evaluated at x = 1?

• a.13.6
• b.5.44
• c.2.72
• d.8.15✓

By the product rule f'(x) = 2x·e^x + x^2·e^x = e^x(2x + x^2). At x = 1, f'(1) = e(2 + 1) = 3e = 8.15. The product rule combines both factors' derivatives.

What is the derivative of f(x) = sin(3x) evaluated at x = 0?

• a.1
• b.0
• c.3✓
• d.-3

By the chain rule d/dx[sin(3x)] = 3cos(3x). At x = 0 this is 3·cos(0) = 3·1 = 3. The multiplier 3 is the derivative of the inner function.

What is the second derivative of f(x) = x^4 evaluated at x = 1?

• a.24
• b.48
• c.4
• d.12✓

The first derivative is 4x^3 and the second derivative is 12x^2. At x = 1, f''(1) = 12(1) = 12. Each differentiation lowers the exponent and multiplies by it.

What is the limit of sin(x)/x as x approaches 0?

• a.Infinity
• b.Undefined
• c.1✓
• d.0

This is a fundamental limit: as x approaches 0, sin(x)/x approaches 1. Although the expression is 0/0 at x = 0, the limit exists and equals 1, the basis of the small-angle approximation.

What is the limit of (1 - cos(x))/x^2 as x approaches 0?

• a.2
• b.1
• c.0.5✓
• d.0

Applying L'Hopital's rule twice (or the Taylor expansion cos(x) approximately 1 - x^2/2) gives the limit 1/2. The numerator behaves like x^2/2 near zero, cancelling the x^2 denominator.

Using the Maclaurin series through the second-order term, estimate e^0.1 (using 1 + x + x^2/2).

• a.0.905
• b.1.105✓
• c.1.2
• d.1.1

The Taylor/Maclaurin series for e^x is 1 + x + x^2/2 + ... At x = 0.1: 1 + 0.1 + (0.01)/2 = 1 + 0.1 + 0.005 = 1.105, very close to the exact 1.10517.

What is the determinant of the matrix [[1, 2, 3], [0, 1, 4], [5, 6, 0]]?

• a.25
• b.1✓
• c.-1
• d.0

Expanding along the first row: 1(1·0 - 4·6) - 2(0·0 - 4·5) + 3(0·6 - 1·5) = 1(-24) - 2(-20) + 3(-5) = -24 + 40 - 15 = 1. Cofactor expansion computes 3x3 determinants.

What are the eigenvalues of the matrix [[4, 1], [2, 3]]?

• a.5 and 2✓
• b.1 and 6
• c.4 and 3
• d.7 and 0

Solve det([[4-lambda, 1], [2, 3-lambda]]) = (4-lambda)(3-lambda) - 2 = lambda^2 - 7·lambda + 10 = 0. Factoring gives lambda = 5 and lambda = 2. The eigenvalues sum to the trace (7) and multiply to the determinant (10).

What is the magnitude of the cross product of vectors (2, 0, 0) and (0, 3, 0)?

• a.5
• b.0
• c.36
• d.6✓

The cross product (2,0,0) x (0,3,0) = (0, 0, 6), whose magnitude is 6. Equivalently, for perpendicular vectors |A x B| = |A||B|sin(90) = 2·3·1 = 6, the area of the spanned parallelogram.

What is the dot product of vectors (1, 2, 3) and (4, -1, 2)?

• a.8✓
• b.0
• c.-2
• d.12

The dot product is the sum of component products: (1)(4) + (2)(-1) + (3)(2) = 4 - 2 + 6 = 8. A positive dot product means the vectors point in generally the same direction.

What is the magnitude of the vector (3, 4, 12)?

• a.13✓
• b.12
• c.19
• d.169

The magnitude is sqrt(3^2 + 4^2 + 12^2) = sqrt(9 + 16 + 144) = sqrt(169) = 13. This extends the Pythagorean theorem to three dimensions.

Solve the system 2x + y = 5 and x - y = 1 for x.

• a.1
• b.2✓
• c.-2
• d.3

Adding the two equations eliminates y: 3x = 6, so x = 2 (and y = 1). Elimination adds equations to cancel a variable.

What are the roots of x^2 - 5x + 6 = 0?

• a.2 and 3✓
• b.1 and 6
• c.-2 and -3
• d.5 and 1

Factoring gives (x - 2)(x - 3) = 0, so the roots are 2 and 3. Their sum equals 5 (the negative of the middle coefficient) and their product equals 6 (the constant term).

What is the magnitude (modulus) of the complex number 3 + 4i?

• a.7
• b.25
• c.3.5
• d.5✓

The modulus is |a + bi| = sqrt(a^2 + b^2) = sqrt(3^2 + 4^2) = sqrt(25) = 5. This is the distance from the origin to the point in the complex plane.

Using Euler's identity, what is the value of e^(i·pi)?

• a.-1✓
• b.0
• c.1
• d.i

Euler's formula e^(i·theta) = cos(theta) + i·sin(theta). At theta = pi: cos(pi) + i·sin(pi) = -1 + 0 = -1. This is the famous Euler identity e^(i·pi) + 1 = 0.

What is the value of log base 2 of 32?

• a.6
• b.5✓
• c.4
• d.16

log2(32) asks 2 to what power equals 32. Since 2^5 = 32, the answer is 5. Logarithms invert exponentiation.

The solution to dy/dx = 0.05y with y(0) = 100 is y = 100·e^(0.05x). What is y at x = 10?

• a.164.9✓
• b.182.2
• c.105.1
• d.150

Exponential growth solves dy/dx = ky as y = y0·e^(kx). Here y(10) = 100·e^(0.05·10) = 100·e^0.5 = 100(1.649) = 164.9. Linear extrapolation to 150 ignores compounding.

The differential equation dy/dx = 3x^2 with y(0) = 2 has solution y = x^3 + 2. What is y at x = 2?

• a.12
• b.8
• c.6
• d.10✓

Integrating 3x^2 gives x^3 + C; the condition y(0) = 2 fixes C = 2, so y = x^3 + 2. At x = 2, y = 8 + 2 = 10. The initial condition sets the constant of integration.

What are the roots of the characteristic equation for the ODE y'' - 5y' + 6y = 0?

• a.r = 1 and r = 6
• b.r = 5 and r = 6
• c.r = 2 and r = 3✓
• d.r = -2 and r = -3

The characteristic equation is r^2 - 5r + 6 = 0, which factors as (r - 2)(r - 3) = 0, giving r = 2 and r = 3. The general solution is y = C1·e^(2x) + C2·e^(3x).

Evaluate the definite integral of x·e^x from 0 to 1.

• a.2.718
• b.1.718
• c.1✓
• d.0.718

Integration by parts gives the antiderivative e^x(x - 1). Evaluating from 0 to 1: [e^1(0)] - [e^0(-1)] = 0 - (-1) = 1. Integration by parts handles products like x·e^x.

What is the average value of f(x) = x^2 on the interval [0, 3]?

• a.1
• b.4.5
• c.9
• d.3✓

The average value is (1/(b-a))·integral of f from a to b = (1/3)·(3^3/3) = (1/3)(9) = 3. The average value spreads the integral evenly across the interval width.

What is the slope of the tangent line to y = x^3 at x = 2?

• a.6
• b.12✓
• c.4
• d.8

The slope of the tangent is the derivative dy/dx = 3x^2. At x = 2, the slope is 3(4) = 12. The value 8 is y itself (2^3), not the slope.

For f(x, y) = x^2·y + y^3, what is the partial derivative with respect to x at the point (1, 2)?

• a.2
• b.4✓
• c.14
• d.5

Treating y as constant, the partial derivative with respect to x is 2xy. At (1, 2) this is 2(1)(2) = 4. The y^3 term vanishes since it has no x.

For f(x, y) = x^2 + y^2, what is the magnitude of the gradient at the point (3, 4)?

• a.25
• b.7
• c.10✓
• d.14

The gradient is (df/dx, df/dy) = (2x, 2y) = (6, 8) at (3, 4). Its magnitude is sqrt(6^2 + 8^2) = sqrt(100) = 10. The gradient points in the direction of steepest increase.

Using the double-angle identity, what is the value of sin(2 x 30 degrees)?

• a.0.577
• b.0.866✓
• c.1
• d.0.5

sin(2 x 30) = sin(60 degrees) = sqrt(3)/2 = 0.866. Equivalently, sin(2·theta) = 2·sin(theta)·cos(theta) = 2(0.5)(0.866) = 0.866.

In a triangle with sides a = 3 and b = 4 and included angle C = 90 degrees, what is the length of side c?

• a.6
• b.5✓
• c.7
• d.25

By the law of cosines c^2 = a^2 + b^2 - 2ab·cos(C). With C = 90, cos(C) = 0, so c^2 = 9 + 16 = 25 and c = 5. This reduces to the Pythagorean theorem for a right angle.

What is the sum of the arithmetic series 1 + 2 + 3 + ... + 100?

• a.5,000
• b.10,100
• c.5,100
• d.5,050✓

The sum of the first n integers is n(n + 1)/2 = 100(101)/2 = 5,050. This is Gauss's formula for an arithmetic series.

What is the sum of the infinite geometric series 1 + 1/2 + 1/4 + 1/8 + ...?

• a.0.5
• b.2✓
• c.1.5
• d.Infinity

For an infinite geometric series with |r| < 1, the sum is a/(1 - r) = 1/(1 - 0.5) = 1/0.5 = 2. Convergence requires the common ratio's magnitude to be below 1.

A point has polar coordinates r = 2, theta = 60 degrees. What is its x-coordinate?

• a.1✓
• b.1.732
• c.0.5
• d.2

The conversion is x = r·cos(theta) = 2·cos(60) = 2(0.5) = 1. The y-coordinate would be r·sin(theta) = 2(0.866) = 1.732.

Using the trapezoidal rule with points f(0) = 1, f(1) = 2, f(2) = 5 and step h = 1, estimate the integral of f from 0 to 2.

• a.4
• b.6
• c.5✓
• d.8

The trapezoidal rule gives h[(f0 + f_n)/2 + sum of interior points] = 1[(1 + 5)/2 + 2] = 1[3 + 2] = 5. Interior points get full weight; endpoints get half weight.