Which of these is NOT a correct basic derivative rule?
d/dx (ln x) = 1/x
d/dx (sin x) = cos x
d/dx (e^x) = x*e^(x-1)
✓
d/dx (x^n) = n*x^(n-1)
Correct Answer
d/dx (e^x) = x*e^(x-1)
The shared property is that the statement provides a correct basic derivative rule. The derivatives of x^n, sin x, and ln x are correctly stated. The derivative of e^x is e^x, not x*e^(x-1).
Question 2
Which of these is NOT a formula for the linear approximation of f(x) near x=a?
f(x) ~= f(x_0) + f'(x_0)(x-x_0)
dy = f'(x) dx
✓
L(x) = f(a) + f'(a)(x-a)
f(x) ~= f(a) + f'(a)(x-a)
Correct Answer
dy = f'(x) dx
The shared property is that the expression represents a formula for the linear approximation of f(x) near a point 'a' (or x_0). The formulas L(x) = f(a) + f'(a)(x-a), f(x) f(a) + f'(a)(x-a), and f(x) f(x_0) + f'(x_0)(x-x_0) are all forms of linear approximation. The expression dy = f'(x) dx represents the differential of y, which is related but not the linear approximation function itself.
Question 3
Which of these properties is NOT generally true for definite integrals?
integral from a to b of c*f(x) dx = c * integral from a to b of f(x) dx
integral from a to b of f(x)*g(x) dx = (integral from a to b of f(x) dx) * (integral from a to b of g(x) dx)
✓
integral from a to b of [f(x) + g(x)] dx = integral from a to b of f(x) dx + integral from a to b of g(x) dx
integral from a to b of f(x) dx = - integral from b to a of f(x) dx
Correct Answer
integral from a to b of f(x)*g(x) dx = (integral from a to b of f(x) dx) * (integral from a to b of g(x) dx)
The shared property is that the statement represents a generally true property of definite integrals. The sum rule, constant multiple rule, and the property _a^b f(x) dx = - _b^a f(x) dx are all true for definite integrals. The statement _a^b f(x) * g(x) dx = _a^b f(x) dx * _a^b g(x) dx is not generally true; there is no product rule for definite integrals in this form.
Question 4
Which of these functions does NOT have a removable discontinuity at x=2?
1 / (x - 2)
✓
(x - 2) / (x^2 - 4)
(x^2 - 3x + 2) / (x - 2)
(x^2 - 4) / (x - 2)
Correct Answer
1 / (x - 2)
The shared property is that the function has a removable discontinuity at x=2. The functions (x^2 - 4) / (x - 2), (x - 2) / (x^2 - 4), and (x^2 - 3x + 2) / (x - 2) all have a limit at x=2 but are undefined at x=2, indicating a removable discontinuity. The function 1 / (x - 2) has an infinite discontinuity at x=2.
Question 5
Which of these differentiation techniques is NOT primarily used for differentiating a composite function, f(g(x))?
Product Rule
✓
Chain Rule
Logarithmic Differentiation
Implicit Differentiation
Correct Answer
Product Rule
The shared property is that the technique is primarily used for differentiating a composite function, f(g(x)). The Chain Rule, Implicit Differentiation (as it applies to functions of y with respect to x), and Logarithmic Differentiation (which often simplifies compositions involving products/quotients/powers) are all techniques heavily reliant on or applied to composite functions. The Product Rule is specifically for differentiating the product of two functions, not a composition.
Question 6
Which of these concepts is NOT primarily determined by analyzing the second derivative of a function?
Inflection Points
Intervals of increasing/decreasing
✓
Second Derivative Test for local extrema
Concavity
Correct Answer
Intervals of increasing/decreasing
The shared property is that the concept is primarily determined by analyzing the second derivative of a function. Concavity, inflection points, and the Second Derivative Test for local extrema all rely on the sign or value of the second derivative. Intervals of increasing/decreasing are determined by the sign of the first derivative.
Question 7
Three of these represent a definite integral. Which one does not?
F(b) - F(a)
integral from a to b of f(x) dx
integral of f(x) dx
✓
lim (n -> infinity) of the sum of f(x_i*) delta x
Correct Answer
integral of f(x) dx
The shared property is that the expression represents or is directly associated with a definite integral. The notation _a^b f(x) dx, the Fundamental Theorem of Calculus Part 2 (F(b) - F(a)), and the Riemann Sum definition (lim (n ) f(x_i*) x) all represent definite integrals. The notation f(x) dx represents an indefinite integral (antiderivative).
Question 8
Which of these limits, when evaluated directly, does NOT result in an indeterminate form suitable for L'Hopital's Rule?
lim (x -> 1) (x^2 - 1) / (x - 1)
lim (x -> 0) 1/x
✓
lim (x -> 0) sin(x)/x
lim (x -> infinity) x^2 / e^x
Correct Answer
lim (x -> 0) 1/x
The shared property is that evaluating the limit directly results in an indeterminate form (0/0 or / ) suitable for L'Hopital's Rule. The limits lim (x 0) sin(x)/x (0/0), lim (x ) x^2 / e^x ( / ), and lim (x 1) (x^2 - 1) / (x - 1) (0/0) all fit this description. The limit lim (x 0) 1/x results in an infinite limit (does not exist), which is not an indeterminate form for L'Hopital's Rule.
Question 9
Which of these functions does NOT have a removable discontinuity at x=0?
f(x) = (e^x - 1) / x
f(x) = (sin x) / x
f(x) = (x^2 + x) / x
f(x) = 1/x
✓
Correct Answer
f(x) = 1/x
The shared property is that the function has a removable discontinuity at x=0. The functions (x^2 + x) / x, (sin x) / x, and (e^x - 1) / x all have a limit at x=0 but are undefined at x=0, indicating a removable discontinuity. The function 1/x has an infinite discontinuity at x=0.
Question 10
Which of these functions does not require the Product Rule to find its derivative?
f(x) = x^2 * sin(x)
f(x) = (x^3 + 1) * cos(x)
f(x) = (x^2 + 5x) / x
✓
f(x) = e^x * ln(x)
Correct Answer
f(x) = (x^2 + 5x) / x
The shared property is that the function requires the Product Rule for its most straightforward differentiation. The functions x^2 * sin(x), e^x * ln(x), and (x^3 + 1) * cos(x) are all products of two distinct functions that must be differentiated using the Product Rule. The function (x^2 + 5x) / x can be simplified to x + 5 before differentiating, which then only requires the Power Rule and constant rule, making the Product Rule unnecessary.
Question 11
Which of these functions does NOT require the application of the Chain Rule for its derivative?
f(x) = sin(x^2)
f(x) = e^(2x)
f(x) = x^4 * cos(x)
✓
f(x) = (x^3 + 1)^5
Correct Answer
f(x) = x^4 * cos(x)
The shared property is that finding the derivative of the function requires the application of the Chain Rule. The functions sin(x^2), (x^3 + 1)^5, and e^(2x) are all composite functions, necessitating the Chain Rule. The function x^4 * cos(x) is a product of two functions, requiring the Product Rule for differentiation.
Question 12
Which of these points on a function's graph is NOT necessarily a critical point?
A local maximum
A point where the derivative is undefined
A local minimum
An inflection point
✓
Correct Answer
An inflection point
The shared property is that the point is necessarily a critical point of the function. A local maximum, a local minimum, and a point where the derivative is undefined (if in the domain) are all by definition critical points. An inflection point is where the concavity changes, and while it might coincide with a critical point, it is not necessarily one.
Question 13
Which of these functions' antiderivatives is NOT found directly using the Power Rule for integration?
f(x) = 1/x
✓
f(x) = x^3
f(x) = sqrt(x)
f(x) = 1/x^2
Correct Answer
f(x) = 1/x
The shared property is that the antiderivative of the function can be found directly using the Power Rule for integration ( x^n dx = x^(n+1)/(n+1) + C, for n -1). The functions x^3, 1/x^2 (x^-2), and sqrt(x) (x^(1/2)) all fit this rule. The function 1/x is a special case whose antiderivative is ln|x| + C, which is not obtained by the Power Rule.
Question 14
Which of these equations does NOT typically require implicit differentiation to find dy/dx?
xy = 1
x^2 + y^2 = 25
e^y + x = y
y = sin(x^2)
✓
Correct Answer
y = sin(x^2)
The shared property is that the equation typically requires implicit differentiation to find dy/dx because y is not easily expressed as an explicit function of x. The equations x^2 + y^2 = 25, xy = 1, and e^y + x = y all require implicit differentiation. The equation y = sin(x^2) explicitly defines y as a function of x, so its derivative can be found directly using the Chain Rule.
Question 15
Which of these scenarios is NOT a typical application of related rates?
Finding the maximum volume of a box made from a sheet of cardboard by cutting squares from the corners.
✓
Finding how fast the shadow of a person is lengthening when they walk away from a lamppost.
Finding how fast the water level in a conical tank is rising when water is being poured in.
Finding how fast the distance between two moving cars is changing.
Correct Answer
Finding the maximum volume of a box made from a sheet of cardboard by cutting squares from the corners.
The shared property is that the scenario is a typical application of related rates, involving finding the rate of change of one quantity in terms of the rate of change of another. The scenarios involving water level in a conical tank, the lengthening shadow, and the changing distance between cars all involve related rates. Finding the maximum volume of a box is an optimization problem, which uses derivatives but not typically in the context of related rates.