Calculus Quiz
Questions: 16 · 10 minutes
1. A particle’s position is s(t) = t³ − 6t² + 9t. What is its instantaneous velocity at t = 2?
−3 units per unit of time, from evaluating s′(2)
0 units per unit of time, treating t = 2 as a turning time
−6 units per unit of time, from differentiating only the middle term
3 units per unit of time, with the direction reversed
2. The limit lim(h→0) [((3 + h)² − 9)/h] represents the derivative of x² at x = 3. What is its value?
0
3
9
6
3. For f(x) = x⁴ − 4x², what does the second derivative indicate about the graph at x = 0?
The graph is concave down there.
The graph is concave up there.
The graph has an inflection point there.
The graph has zero concavity there.
4. A rectangle has a perimeter of 20 units. Which dimensions give it the greatest possible area?
1 unit by 9 units
2 units by 8 units
5 units by 5 units
4 units by 6 units
5. The equation x² + y² = 25 defines a circle. What is dy/dx at the point (3, 4)?
A positive slope of 3/4
A negative slope of −3/4
A negative slope of −4/3
A positive slope of 4/3
6. A cost model is C(q) = q⁵. What is the marginal cost C′(q) when q = 2?
32, obtained by evaluating the original cost model
160, obtained by multiplying the cost by the exponent
80, obtained from C′(q) = 5q⁴
64, obtained from the rule C′(q) = q⁶/6
7. A student needs the critical numbers of f(x) = x³ − 3x. Which values should they report?
x = 0 only
x = −1 and x = 1
x = −3 and x = 3
x = 1 only
8. If g(x) = sin(3x), what is g′(x)?
cos(3x)
sin(3x)/3
−3sin(3x)
3cos(3x)
9. A circular ripple expands so that its radius increases at 1 unit per second. How fast is its area increasing when the radius is 5 units?
5π square units per second
25π square units per second
10π square units per second
2π square units per second
10. For f(x) = x² on [1, 3], the Mean Value Theorem guarantees a point c where the instantaneous rate equals the average rate over the interval. What is c?
c = 1
c = 2
c = 5/2
c = 3/2
11. Suppose f is differentiable and invertible, with f(2) = 5 and f′(2) = 3. What is (f⁻¹)′(5)?
3
1/3
1/5
2/5
12. For x > 0, what is the derivative of f(x) = ln(x)/x?
1/x²
(ln(x) − 1)/x²
(1 − ln(x))/x²
1 − ln(x)
13. The curve y = x² has a tangent at x = 2. Which line has the correct slope and passes through the point of tangency?
y = 4x − 4, with slope 4 through (2, 4)
y = 2x, with slope 2 through (2, 4)
y = 4x + 4, with slope 4 but the wrong intercept
y = 2x + 2, passing through (2, 6) instead
14. Which statement correctly describes f(x) = |x| at x = 0?
It is differentiable but not continuous there.
It is both continuous and differentiable there.
It is neither continuous nor differentiable there.
It is continuous but not differentiable there.
15. Use the product rule to differentiate f(x) = x²eˣ. Which expression results?
eˣ(x² + 2x)
x²eˣ
2xeˣ
2xeˣ + x²
16. Use the linearization of f(x) = √x at x = 9 to estimate √9.3. What estimate results?
3.30, from adding the full change 0.3 to √9
3.10, from using a slope of 1/3 at x = 9
3.02, from dividing the change by 18
3.05, from using L(x) = 3 + (x − 9)/6