Mathematics practice questions

Answer: B) 2

The antiderivative of 1/x is ln x, so the integral from 1 to e^2 is ln(e^2) − ln(1) = 2.

• AIncorrect. This is just the upper limit, with no antiderivative applied. The antiderivative of 1/x is ln x, so the result is ln(e^2) − ln 1.
• BCorrect. ln(e^2) − ln(1) = 2 − 0 = 2.
• CIncorrect. 1 is the value of the integral from 1 to e. With an upper limit of e^2, the result is ln(e^2) = 2.
• DIncorrect. This treats the integrand as the constant 1, which gives the width of the interval. The integrand is 1/x, whose antiderivative is ln x.

Answer: B) 19

The slope is the first derivative: dy/dx = 6x² − 5. At x = 2 this gives 24 − 5 = 19.

• AIncorrect. 6 is the value of the function at x = 2 (16 − 10). It is not the derivative.
• BCorrect. dy/dx = 6x² − 5, which at x = 2 is 6(4) − 5 = 19.
• CIncorrect. 24 comes from dropping the derivative of −5x, leaving 6x² = 24. It also equals the second derivative 12x at x = 2.
• DIncorrect. 7 comes from misapplying the power rule to get 6x − 5. The derivative of 2x³ is 6x², not 6x.

Answer: A) 2.025

• ACorrect. \(2 + 0.1/(2\sqrt{4}) = 2 + 0.025\).
• BIncorrect. The derivative is \(1/(2\sqrt{x}) = 0.25\).
• CIncorrect. Multiply 0.1 by the derivative, 0.25.
• DIncorrect. Recheck: 0.1 × 0.25.

Answer: B) 3

• AIncorrect. That's the run; divide rise by run.
• BCorrect. (7 − 1)/(3 − 1) = 6/2.
• CIncorrect. That's the rise; divide by the run.
• DIncorrect. That's run over rise, inverted.

Answer: B) 2 and 5

Find two numbers that add to 7 and multiply to 10. That gives (x − 2)(x − 5) = 0, so x = 2 or x = 5.

• AIncorrect. These are the coefficients of the equation, not its roots. Substituting x = 7 gives 49 − 49 + 10 = 10, which is not 0.
• BCorrect. The quadratic factors as (x − 2)(x − 5) = 0. The roots add to 7 and multiply to 10, as required.
• CIncorrect. Their product is 10, but their sum is 11, not 7.
• DIncorrect. Their sum is 7, but their product is 12, not 10.

Answer: C) 45°

• AIncorrect. tan 60° ≈ 1.732.
• BIncorrect. tan 30° ≈ 0.577.
• CCorrect. tan 45° = 1.
• DIncorrect. tan 90° is undefined.

Answer: D) \(x = e^2 \approx 7.39\)

• AIncorrect. That's \(\ln 2\).
• BIncorrect. That ignores the logarithm.
• CIncorrect. That's for \(\log_{10} x = 2\).
• DCorrect. Exponentiate both sides.

Answer: B) 6.28 m

• AIncorrect. That's 360°.
• BCorrect. \(s = r\theta = 4 \times \pi/2\).
• CIncorrect. That's 180°.
• DIncorrect. That uses a 2-m radius.

Answer: C) A repeated root at x = 3

• AIncorrect. The discriminant is zero.
• BIncorrect. Those don't satisfy the equation.
• CCorrect. It factors as \((x - 3)^2\).
• DIncorrect. Those don't satisfy the equation.

Answer: A) 4

Average rate of change is the secant slope [f(b) − f(a)]/(b − a) = (9 − 1)/(3 − 1) = 4.

• ACorrect. [f(3) − f(1)]/(3 − 1) = (9 − 1)/2 = 4.
• BIncorrect. 8 is only the change in f; it must be divided by the change in x (2).
• CIncorrect. 6 is f′(3) = 2(3), the instantaneous rate at x = 3, not the average over the interval.
• DIncorrect. 5 is the average of the function values (1 + 9)/2, not the rate of change of f per unit x.