If f(x)=5x^4-2x+7, then f'(x) equals:
- 20x^3-2 Correct answer.
- 5x^3-2
- 20x^3-2x
- 20x^4-2
Explanation
Differentiate term by term: d/dx[5x^4] = 20x^3, d/dx[-2x] = -2, and the constant 7 contributes 0. So f'(x) = 20x^3 - 2.
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If f(x)=e^{2x}, then f'(x) equals:
- 2e^{2x} Correct answer.
- e^{2x}
- 2e^x
- e^{2x}/2
Explanation
Chain rule: d/dx[e^{u}] = e^{u} · u' with u = 2x. Thus f'(x) = e^{2x} · 2 = 2e^{2x}.
For f(x)=x^2-4x+1, the critical point f'(x)=0 is at x=
- 2 Correct answer.
- 4
- 1
- 0
Explanation
Compute f'(x) = 2x - 4. Set the derivative equal to zero: 2x - 4 = 0 ⇒ x = 2. That is the only critical point.
Slope of the tangent to y=x^2 at x=3 is:
- 6 Correct answer.
- 9
- 3
- 12
Explanation
Differentiate y = x^2 to get y' = 2x. Evaluating at x = 3 gives slope 6 for the tangent line.
d/dx[x sin x] equals:
- sin x + x cos x Correct answer.
- x cos x
- sin x + cos x
- cos x - x sin x
Explanation
Product rule (uv)' = u'v + uv' with u = x and v = sin x: u' = 1, v' = cos x. Therefore the derivative is sin x + x cos x.
d/dx[(x^2)/(x+1)] equals:
- (x^2+2x)/(x+1)^2 Correct answer.
- 2x/(x+1)
- x^2/(x+1)^2
- (2x(x+1)-x^2)/(x+1)
Explanation
Quotient rule: (num' den - num den')/den^2. With num = x^2 and den = x + 1: (2x(x + 1) - x^2 · 1)/(x + 1)^2 = (2x^2 + 2x - x^2)/(x + 1)^2 = (x^2 + 2x)/(x + 1)^2.
If f(x)=(3x+1)^5, then f'(x) equals:
- 15(3x+1)^4 Correct answer.
- 5(3x+1)^4
- 15(3x+1)^5
- 3(3x+1)^4
Explanation
Chain rule for (g(x))^n: n(g(x))^{n-1} g'(x). Here g(x) = 3x + 1, so f'(x) = 5(3x + 1)^4 · 3 = 15(3x + 1)^4.
f(x)=x^3-3x has a local maximum at x=
- -1 Correct answer.
- 1
- 0
- 3
Explanation
f'(x) = 3x^2 - 3 = 3(x - 1)(x + 1) = 0 gives critical points x = ±1. Second derivative f''(x) = 6x: f''(-1) = -6 < 0 (local max), f''(1) = 6 > 0 (local min). So the local maximum is at x = -1.
Tangent line to y=x^2 at (1,1) is:
- y=2x-1 Correct answer.
- y=2x+1
- y=x
- y=2x
Explanation
Slope of y = x^2 is y' = 2x; at x = 1 the slope is 2. Point-slope form through (1, 1): y - 1 = 2(x - 1) ⇒ y = 2x - 1.
If y=ln(3x), then dy/dx equals:
- 1/x Correct answer.
- 3/x
- 1/(3x)
- 3 ln(3x)
Explanation
Rewrite ln(3x) = ln 3 + ln x for x > 0. Differentiating gives 0 + 1/x = 1/x. Chain rule confirms: (1/(3x)) · 3 = 1/x.
d/dx[sin(2x)] equals:
- 2 cos(2x) Correct answer.
- cos(2x)
- 2 sin(2x)
- -2 cos(2x)
Explanation
Chain rule: d/dx[sin(u)] = cos(u) · u' with u = 2x. Thus the derivative is cos(2x) · 2 = 2 cos(2x).
Position s(t)=t^3-6t^2+9t. Velocity at t=2 is:
- -3 Correct answer.
- 0
- 3
- -6
Explanation
Velocity is the derivative of position: v(t) = 3t^2 - 12t + 9. Substituting t = 2: v(2) = 3·4 - 12·2 + 9 = 12 - 24 + 9 = -3.
Linear approximation: estimate √4.1 near a=4. Result:
- 2.025 Correct answer.
- 2.05
- 2.1
- 2.01
Explanation
Linearization of f(x) = √x at a = 4: f(4) = 2 and f'(x) = 1/(2√x), so f'(4) = 1/4. Then f(4.1) ≈ 2 + (0.1)(1/4) = 2.025.
For f(x)=x e^{-x}, a critical point is at x=
- 1 Correct answer.
- 0
- -1
- e
Explanation
Product rule: f'(x) = e^{-x} + x(-e^{-x}) = e^{-x}(1 - x). Since e^{-x} never vanishes, f'(x) = 0 precisely when 1 - x = 0, i.e. x = 1.
Implicit: x^2+y^2=25. At (3,4), dy/dx equals:
- -3/4 Correct answer.
- 3/4
- -4/3
- 4/3
Explanation
Differentiate implicitly: 2x + 2y y' = 0 ⇒ y' = -x/y. At the point (3, 4) on the circle, y' = -3/4.
∫ 2x dx equals (plus C):
- x^2 + C Correct answer.
- 2x^2 + C
- x + C
- 2 + C
Explanation
Reverse the power rule: an antiderivative of 2x is x^2. Including the arbitrary constant gives x^2 + C.
∫ x^3 dx equals (plus C):
- x^4/4 + C Correct answer.
- x^4 + C
- 3x^2 + C
- x^3/3 + C
Explanation
For n ≠ -1, ∫ x^n dx = x^{n+1}/(n + 1) + C. With n = 3 this is x^4/4 + C.
∫ e^x dx equals (plus C):
- e^x + C Correct answer.
- x e^x + C
- e^{x+1} + C
- ln|e^x| + C
Explanation
The function e^x is its own antiderivative, so ∫ e^x dx = e^x + C.
∫ (1/x) dx for x>0 equals (plus C):
- ln x + C Correct answer.
- 1/x^2 + C
- x ln x + C
- 1/x + C
Explanation
The standard integral formula is ∫ (1/x) dx = ln|x| + C. Restricting to x > 0 gives ln x + C.
∫_0^2 3x^2 dx equals:
- 8 Correct answer.
- 6
- 12
- 4
Explanation
An antiderivative of 3x^2 is x^3. Evaluating the definite integral: [x^3]_0^2 = 8 - 0 = 8.
∫_0^{π/2} cos x dx equals:
- 1 Correct answer.
- 0
- 2
- π/2
Explanation
An antiderivative of cos x is sin x. Evaluating from 0 to π/2: sin(π/2) - sin(0) = 1 - 0 = 1.
∫ sin x dx equals (plus C):
- -cos x + C Correct answer.
- cos x + C
- sin x + C
- -sin x + C
Explanation
Because d/dx[-cos x] = sin x, the indefinite integral is ∫ sin x dx = -cos x + C.
By substitution, ∫ 2x(x^2+1)^3 dx equals (plus C):
- (x^2+1)^4/4 + C Correct answer.
- (x^2+1)^4 + C
- 2(x^2+1)^3 + C
- (x^2+1)^3/3 + C
Explanation
Substitute u = x^2 + 1, so du = 2x dx. The integral becomes ∫ u^3 du = u^4/4 + C = (x^2 + 1)^4/4 + C.
Area under y=x from 0 to 4 equals:
- 8 Correct answer.
- 4
- 16
- 2
Explanation
Compute ∫_0^4 x dx = [x^2/2]_0^4 = 8. Geometrically this is a right triangle with base 4 and height 4, so area = (1/2)·4·4 = 8.
∫_1^e (1/x) dx equals:
- 1 Correct answer.
- e
- 0
- e-1
Explanation
∫ (1/x) dx = ln|x|. From 1 to e: ln e - ln 1 = 1 - 0 = 1.
If F'=f, F(0)=2, F(3)=8, then ∫_0^3 f dx equals:
- 6 Correct answer.
- 8
- 2
- 10
Explanation
By the Fundamental Theorem of Calculus, ∫_a^b f(x) dx = F(b) - F(a) when F' = f. Here ∫_0^3 f = F(3) - F(0) = 8 - 2 = 6.
∫_0^1 (2x+1) dx equals:
- 2 Correct answer.
- 1
- 3
- 1.5
Explanation
An antiderivative of 2x + 1 is x^2 + x. Evaluating from 0 to 1: (1 + 1) - (0 + 0) = 2.
Integration by parts: ∫ x e^x dx equals (plus C):
- e^x(x-1)+C Correct answer.
- x e^x + C
- e^x(x+1)+C
- x^2 e^x/2 + C
Explanation
Integration by parts: ∫ u dv = uv - ∫ v du. Set u = x, dv = e^x dx, so du = dx, v = e^x. Then ∫ x e^x dx = x e^x - ∫ e^x dx = x e^x - e^x + C = e^x(x - 1) + C.
Average value of f(x)=x^2 on [0,3] is:
- 3 Correct answer.
- 9
- 1
- 27/2
Explanation
Average value on [a, b] is (1/(b - a)) ∫_a^b f. Here (1/3) ∫_0^3 x^2 dx = (1/3)[x^3/3]_0^3 = (1/3)·9 = 3.
Improper integral ∫_1^∞ x^{-2} dx equals:
- 1 Correct answer.
- ∞
- 0
- 1/2
Explanation
Evaluate the limit: lim_{b→∞} ∫_1^b x^{-2} dx = lim_{b→∞} [-1/x]_1^b = lim_{b→∞}(-1/b + 1) = 1. The improper integral converges to 1.
Let f(x) = 3x^2 - 5x + 2. What is f(2)?
- 4 Correct answer.
- 6
- 8
- 12
Explanation
Substitute x = 2: f(2) = 3(2)^2 - 5(2) + 2 = 3·4 - 10 + 2 = 12 - 10 + 2 = 4.
The quadratic f(x) = x^2 - 6x + 5 has roots at:
- x = 1 and x = 5 Correct answer.
- x = -1 and x = -5
- x = 2 and x = 3
- x = 0 and x = 6
Explanation
Factor x^2 - 6x + 5 = (x - 1)(x - 5). Setting each factor to zero gives roots x = 1 and x = 5. The quadratic formula with Δ = 36 - 20 = 16 yields the same roots: (6 ± 4)/2.
What is the vertex of the parabola y = x^2 - 4x + 7?
- (2, 3) Correct answer.
- (2, 7)
- (-2, 3)
- (4, 7)
Explanation
For y = ax^2 + bx + c the vertex abscissa is x = -b/(2a) = 4/2 = 2. Then y = 2^2 - 4·2 + 7 = 4 - 8 + 7 = 3. Completing the square gives y = (x - 2)^2 + 3, confirming vertex (2, 3).
The domain of f(x) = 1/(x - 3) is all real x except:
- x = 3 Correct answer.
- x = 0
- x = 1
- x = -3
Explanation
A rational function is undefined where the denominator is zero. Solving x - 3 = 0 gives the excluded value x = 3. Every other real number is in the domain.
Simplify (x^2 - 9)/(x - 3) for x ≠ 3.
- x + 3 Correct answer.
- x - 3
- x^2 - 3
- 9
Explanation
Difference of squares: x^2 - 9 = (x - 3)(x + 3). Cancelling the common factor (x - 3) for x ≠ 3 leaves x + 3.
log_2(16) equals:
- 4 Correct answer.
- 2
- 8
- 16
Explanation
log_2(16) is the exponent that produces 16 when the base is 2. Since 2^4 = 16, log_2(16) = 4.
The range of f(x) = e^x is:
- (0, ∞) Correct answer.
- (-∞, ∞)
- [0, ∞)
- (-∞, 0)
Explanation
The exponential e^x is positive for every real x. As x → -∞, e^x → 0 but never reaches 0; as x → ∞, e^x → ∞. Therefore the range is the open interval (0, ∞).
If f(x) = (x - 1)/(x + 2), what is the horizontal asymptote?
- y = 1 Correct answer.
- y = -2
- y = 0
- y = -1
Explanation
For a rational function with equal numerator and denominator degrees, the horizontal asymptote is the ratio of leading coefficients. Here both leading coefficients are 1, so y = 1.
The function f(x) = log_3(x - 1) is defined for:
- x > 1 Correct answer.
- x ≥ 1
- x > 0
- x ≥ 0
Explanation
A real logarithm is defined only for positive arguments. Require x - 1 > 0, so x > 1.
Solve x^2 = 49. The solutions are:
- x = 7 or x = -7 Correct answer.
- x = 7 only
- x = 49
- x = √49 only
Explanation
Taking square roots of both sides of x^2 = 49 gives x = ±√49 = ±7. Both values satisfy the original equation.
The solution set of 3x + 1 > 10 is:
- x > 3 Correct answer.
- x ≥ 3
- x < 3
- x > 11/3
Explanation
Subtract 1: 3x > 9. Divide by the positive number 3 (inequality direction unchanged): x > 3.
Solve |x - 2| = 5. The solutions are:
- x = 7 or x = -3 Correct answer.
- x = 7 only
- x = 3 or x = -7
- x = 5
Explanation
Solve the two cases: x - 2 = 5 ⇒ x = 7, and x - 2 = -5 ⇒ x = -3. Both satisfy |x - 2| = 5.
Solve 2^{x} = 32. What is x?
- 5 Correct answer.
- 4
- 16
- 6
Explanation
Express 32 as a power of 2: 32 = 2^5. Therefore the equation 2^x = 32 forces x = 5.
Solve log_3(x) = 2. What is x?
- 9 Correct answer.
- 6
- 8
- 3
Explanation
By definition, log_3(x) = 2 means 3^2 = x, so x = 9. Also check: log_3(9) = 2.
The discriminant of x^2 - 4x + 5 = 0 is:
- -4 Correct answer.
- 4
- 0
- 16
Explanation
Discriminant Δ = b^2 - 4ac = (-4)^2 - 4(1)(5) = 16 - 20 = -4. A negative discriminant means no real roots.
Solve the system x + y = 10 and x - y = 4. What is (x, y)?
- (7, 3) Correct answer.
- (6, 4)
- (5, 5)
- (8, 2)
Explanation
Add the equations: (x + y) + (x - y) = 10 + 4 ⇒ 2x = 14 ⇒ x = 7. Then y = 10 - 7 = 3. Check: 7 - 3 = 4.
For which real x is (x - 1)(x + 2) ≤ 0?
- [-2, 1] Correct answer.
- (-2, 1)
- (-∞, -2] ∪ [1, ∞)
- [1, 2]
Explanation
The roots of (x - 1)(x + 2) = 0 are x = 1 and x = -2. A product of two linear factors is ≤ 0 between the roots, inclusive of the roots where the product is zero. Hence x ∈ [-2, 1].
Solve e^{2x} = e^{x+3}. What is x?
- 3 Correct answer.
- 0
- 1
- 3/2
Explanation
The exponential function is one-to-one, so e^{2x} = e^{x+3} implies 2x = x + 3. Subtracting x yields x = 3.
Arithmetic sequence: a_1 = 3, d = 4. What is a_5?
- 19 Correct answer.
- 15
- 20
- 23
Explanation
Arithmetic nth term: a_n = a_1 + (n - 1)d. Here a_5 = 3 + (5 - 1)·4 = 3 + 16 = 19.
Geometric sequence: a_1 = 2, r = 3. What is a_4?
- 54 Correct answer.
- 24
- 18
- 162
Explanation
Geometric nth term: a_n = a_1 r^{n-1}. Here a_4 = 2 · 3^{3} = 2 · 27 = 54.