Problem 1
Evaluate \(\displaystyle\lim_{x \to 3} \dfrac{x^2 - 9}{x - 3}\).
A) \(6\)
B) \(0\)
C) \(3\)
D) \(9\)
E) \(12\)
Solution:
Factor: \(\dfrac{(x-3)(x+3)}{x-3} = x + 3\) for \(x \neq 3\).
Take the limit: \(3 + 3\).
\(= \boxed{6}\)
Problem 2
Evaluate \(\displaystyle\lim_{x \to \infty} \dfrac{3x^2 + 2x}{x^2 - 5}\).
A) \(0\)
B) \(3\)
C) \(1\)
D) \(2\)
E) \(5\)
Solution:
Degrees of numerator and denominator are equal (2).
The limit is the ratio of leading coefficients: \(\dfrac{3}{1}\).
\(= \boxed{3}\)
Problem 3
If \(f(x) = x^3 - 2x^2 + 5x - 1\), find \(f'(2)\).
A) \(5\)
B) \(7\)
C) \(9\)
D) \(11\)
E) \(4\)
Solution:
\(f'(x) = 3x^2 - 4x + 5\).
\(f'(2) = 3(4) - 4(2) + 5 = 12 - 8 + 5\).
\(= \boxed{9}\)
Problem 4
Differentiate \(f(x) = x^2 \sin x\).
A) \(2x\cos x\)
B) \(x^2 \cos x\)
C) \(2x\sin x\)
D) \(2x\sin x + x^2 \cos x\)
E) \(2x\sin x - x^2 \cos x\)
Solution:
Product rule: \((uv)' = u'v + uv'\) with \(u = x^2\), \(v = \sin x\).
\(= (2x)\sin x + x^2(\cos x)\).
\(= \boxed{2x\sin x + x^2 \cos x}\)
Problem 5
Differentiate \(f(x) = \dfrac{x}{x + 1}\).
A) \(\dfrac{1}{x + 1}\)
B) \(\dfrac{x}{(x + 1)^2}\)
C) \(-\dfrac{1}{(x + 1)^2}\)
D) \(\dfrac{1}{x^2 + 1}\)
E) \(\dfrac{1}{(x + 1)^2}\)
Solution:
Quotient rule: \(\dfrac{u'v - uv'}{v^2}\) with \(u = x\), \(v = x+1\).
\(= \dfrac{(1)(x+1) - x(1)}{(x+1)^2} = \dfrac{1}{(x+1)^2}\).
\(= \boxed{\dfrac{1}{(x + 1)^2}}\)
Problem 6
Differentiate \(f(x) = (2x + 1)^5\).
A) \(10(2x + 1)^4\)
B) \(5(2x + 1)^4\)
C) \((2x + 1)^4\)
D) \(10(2x + 1)^5\)
E) \(2(2x + 1)^4\)
Solution:
Chain rule: bring down 5, reduce power, times the inner derivative (2).
\(5(2x+1)^4 \cdot 2\).
\(= \boxed{10(2x + 1)^4}\)
Problem 7
Differentiate \(f(x) = \sin(3x)\).
A) \(\cos(3x)\)
B) \(3\cos(3x)\)
C) \(-3\cos(3x)\)
D) \(3\sin(3x)\)
E) \(-3\sin(3x)\)
Solution:
\(\dfrac{d}{dx}\sin(u) = \cos(u)\cdot u'\) with \(u = 3x\).
\(= \cos(3x)\cdot 3\).
\(= \boxed{3\cos(3x)}\)
Problem 8
Differentiate \(f(x) = e^{2x}\).
A) \(e^{2x}\)
B) \(2e^{x}\)
C) \(2e^{2x}\)
D) \(\dfrac{e^{2x}}{2}\)
E) \(2x\,e^{2x-1}\)
Solution:
\(\dfrac{d}{dx}e^{u} = e^{u}\cdot u'\) with \(u = 2x\).
\(= e^{2x}\cdot 2\).
\(= \boxed{2e^{2x}}\)
Problem 9
Find the slope of the line tangent to \(y = x^2\) at the point \((1, 1)\).
A) \(1\)
B) \(4\)
C) \(\dfrac{1}{2}\)
D) \(2\)
E) \(0\)
Solution:
The slope of the tangent is \(\dfrac{dy}{dx} = 2x\).
At \(x = 1\): \(2(1)\).
\(= \boxed{2}\)
Problem 10
For the circle \(x^2 + y^2 = 25\), find \(\dfrac{dy}{dx}\) at the point \((3, 4)\).
A) \(\dfrac{3}{4}\)
B) \(-\dfrac{4}{3}\)
C) \(\dfrac{4}{3}\)
D) \(-\dfrac{3}{5}\)
E) \(-\dfrac{3}{4}\)
Solution:
Implicit differentiation: \(2x + 2y\,\dfrac{dy}{dx} = 0\).
\(\dfrac{dy}{dx} = -\dfrac{x}{y} = -\dfrac{3}{4}\).
\(= \boxed{-\dfrac{3}{4}}\)
Problem 11
The radius of a circle increases at \(2\) cm/s. How fast is the area increasing when \(r = 5\) cm?
A) \(20\pi\) cm\(^2\)/s
B) \(10\pi\) cm\(^2\)/s
C) \(4\pi\) cm\(^2\)/s
D) \(100\pi\) cm\(^2\)/s
E) \(40\pi\) cm\(^2\)/s
Solution:
\(A = \pi r^2 \Rightarrow \dfrac{dA}{dt} = 2\pi r\,\dfrac{dr}{dt}\).
\(= 2\pi (5)(2)\).
\(= \boxed{20\pi}\) cm\(^2\)/s
Problem 12
At what value of \(x\) does \(f(x) = x^3 - 3x\) have a local maximum?
A) \(x = 1\)
B) \(x = -1\)
C) \(x = 0\)
D) \(x = 3\)
E) \(x = -3\)
Solution:
\(f'(x) = 3x^2 - 3 = 0 \Rightarrow x = \pm 1\).
\(f''(x) = 6x\); at \(x = -1\), \(f'' < 0\) (concave down).
Local maximum at \(\boxed{x = -1}\)
Problem 13
For \(f(x) = x^2\) on \([1, 3]\), find the value \(c\) guaranteed by the Mean Value Theorem.
A) \(4\)
B) \(1\)
C) \(2\)
D) \(\sqrt{3}\)
E) \(3\)
Solution:
MVT: \(f'(c) = \dfrac{f(3) - f(1)}{3 - 1} = \dfrac{9 - 1}{2} = 4\).
\(f'(c) = 2c = 4\).
\(c = \boxed{2}\)
Problem 14
Evaluate \(\displaystyle\int_0^2 (3x^2 + 2)\,dx\).
A) \(8\)
B) \(16\)
C) \(14\)
D) \(12\)
E) \(10\)
Solution:
Antiderivative: \(x^3 + 2x\).
\(\big[x^3 + 2x\big]_0^2 = (8 + 4) - 0\).
\(= \boxed{12}\)
Problem 15
If \(g(x) = \displaystyle\int_0^x (t^2 + 1)\,dt\), find \(g'(2)\).
A) \(4\)
B) \(2\)
C) \(1\)
D) \(3\)
E) \(5\)
Solution:
By the Fundamental Theorem of Calculus, \(g'(x) = x^2 + 1\).
\(g'(2) = 2^2 + 1\).
\(= \boxed{5}\)
Problem 16
Find the area of the region bounded by \(y = x^2\), the \(x\)-axis, and \(x = 2\).
A) \(\dfrac{8}{3}\)
B) \(4\)
C) \(2\)
D) \(\dfrac{16}{3}\)
E) \(8\)
Solution:
Area \(= \displaystyle\int_0^2 x^2\,dx = \left[\dfrac{x^3}{3}\right]_0^2\).
\(= \dfrac{8}{3} - 0\).
\(= \boxed{\dfrac{8}{3}}\)
Problem 17
Evaluate \(\displaystyle\int_0^1 2x(x^2 + 1)^3\,dx\).
A) \(4\)
B) \(\dfrac{15}{4}\)
C) \(15\)
D) \(\dfrac{1}{4}\)
E) \(8\)
Solution:
Let \(u = x^2 + 1\), \(du = 2x\,dx\); bounds \(u: 1 \to 2\).
\(\displaystyle\int_1^2 u^3\,du = \left[\dfrac{u^4}{4}\right]_1^2 = \dfrac{16}{4} - \dfrac{1}{4}\).
\(= \boxed{\dfrac{15}{4}}\)
Problem 18
Find the average value of \(f(x) = x^2\) on \([0, 3]\).
A) \(9\)
B) \(6\)
C) \(3\)
D) \(1\)
E) \(2\)
Solution:
Average value \(= \dfrac{1}{b-a}\displaystyle\int_a^b f(x)\,dx = \dfrac{1}{3}\int_0^3 x^2\,dx\).
\(= \dfrac{1}{3}\left[\dfrac{x^3}{3}\right]_0^3 = \dfrac{1}{3}(9)\).
\(= \boxed{3}\)
Problem 19
A particle moves with position \(s(t) = t^3 - 6t^2 + 9t\). Find the sum of the times \(t \ge 0\) at which the particle is at rest.
A) \(1\)
B) \(3\)
C) \(2\)
D) \(4\)
E) \(6\)
Solution:
Velocity \(v(t) = s'(t) = 3t^2 - 12t + 9 = 3(t-1)(t-3)\).
At rest when \(v = 0\): \(t = 1\) and \(t = 3\).
Sum \(= 1 + 3 = \boxed{4}\)
Problem 20
Approximate \(\displaystyle\int_0^4 x^2\,dx\) using a right Riemann sum with two rectangles of equal width.
A) \(8\)
B) \(16\)
C) \(20\)
D) \(32\)
E) \(40\)
Solution:
Width \(\Delta x = 2\); right endpoints \(x = 2, 4\).
Sum \(= 2\,f(2) + 2\,f(4) = 2(4) + 2(16)\).
\(= 8 + 32 = \boxed{40}\)
Problem 21
Evaluate \(\displaystyle\int_0^1 x e^x\,dx\).
A) \(1\)
B) \(e\)
C) \(e - 1\)
D) \(0\)
E) \(2e\)
Solution:
Integration by parts: \(u = x\), \(dv = e^x dx\); \(\int x e^x dx = e^x(x - 1)\).
\(\big[e^x(x-1)\big]_0^1 = e(0) - (1)(-1)\).
\(= 0 + 1 = \boxed{1}\)
Problem 22
Evaluate the improper integral \(\displaystyle\int_1^{\infty} \dfrac{1}{x^2}\,dx\).
A) \(0\)
B) \(1\)
C) \(2\)
D) \(\dfrac{1}{2}\)
E) \(3\)
Solution:
\(\displaystyle\int_1^{\infty} x^{-2}\,dx = \left[-\dfrac{1}{x}\right]_1^{\infty}\).
\(= 0 - (-1)\).
\(= \boxed{1}\)
Problem 23
Evaluate \(\displaystyle\sum_{n=0}^{\infty} 2\left(\dfrac{1}{3}\right)^n\).
A) \(2\)
B) \(\dfrac{3}{2}\)
C) \(3\)
D) \(6\)
E) \(\dfrac{9}{2}\)
Solution:
Geometric series with \(a = 2\), \(r = \dfrac{1}{3}\).
Sum \(= \dfrac{a}{1 - r} = \dfrac{2}{1 - \tfrac13} = \dfrac{2}{\tfrac23}\).
\(= \boxed{3}\)
Problem 24
In the Maclaurin series for \(e^x\), what is the coefficient of \(x^3\)?
A) \(\dfrac{1}{2}\)
B) \(1\)
C) \(\dfrac{1}{3}\)
D) \(\dfrac{1}{6}\)
E) \(\dfrac{1}{24}\)
Solution:
\(e^x = \displaystyle\sum_{n=0}^{\infty}\dfrac{x^n}{n!}\), so the \(x^3\) coefficient is \(\dfrac{1}{3!}\).
\(3! = 6\).
\(= \boxed{\dfrac{1}{6}}\)
Problem 25
A curve is given parametrically by \(x = t^2\), \(y = t^3\). Find \(\dfrac{dy}{dx}\) at \(t = 2\).
A) \(6\)
B) \(\dfrac{3}{2}\)
C) \(2t\)
D) \(\dfrac{3t}{2}\)
E) \(3\)
Solution:
\(\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt} = \dfrac{3t^2}{2t} = \dfrac{3t}{2}\).
At \(t = 2\): \(\dfrac{3(2)}{2}\).
\(= \boxed{3}\)
Problem 26
Find the area enclosed by the polar curve \(r = 2\cos\theta\).
A) \(\pi\)
B) \(2\pi\)
C) \(4\pi\)
D) \(\dfrac{\pi}{2}\)
E) \(2\)
Solution:
\(r = 2\cos\theta\) is a circle of radius \(1\) (diameter from origin to \((2,0)\)).
Area \(= \pi(1)^2\) (or \(\tfrac12\displaystyle\int_{-\pi/2}^{\pi/2}(2\cos\theta)^2 d\theta = \pi\)).
\(= \boxed{\pi}\)
Problem 27
Find the arc length of \(y = \dfrac{2}{3}x^{3/2}\) from \(x = 0\) to \(x = 3\).
A) \(\dfrac{7}{3}\)
B) \(\dfrac{14}{3}\)
C) \(9\)
D) \(\dfrac{8}{3}\)
E) \(\dfrac{28}{3}\)
Solution:
\(y' = x^{1/2}\), so \(1 + (y')^2 = 1 + x\).
\(L = \displaystyle\int_0^3 \sqrt{1 + x}\,dx = \left[\dfrac{2}{3}(1+x)^{3/2}\right]_0^3 = \dfrac{2}{3}(8 - 1)\).
\(= \boxed{\dfrac{14}{3}}\)
Problem 28
Decompose \(\dfrac{1}{(x-1)(x+1)} = \dfrac{A}{x-1} + \dfrac{B}{x+1}\). Find \(A\).
A) \(1\)
B) \(-\dfrac{1}{2}\)
C) \(\dfrac{1}{2}\)
D) \(2\)
E) \(-1\)
Solution:
Multiply by \((x-1)\) and let \(x = 1\): \(A = \dfrac{1}{x+1}\Big|_{x=1}\).
\(= \dfrac{1}{1 + 1}\).
\(A = \boxed{\dfrac{1}{2}}\)
Problem 29
A population satisfies the logistic equation \(\dfrac{dP}{dt} = 0.5P\left(1 - \dfrac{P}{200}\right)\). What is the carrying capacity?
A) \(0.5\)
B) \(100\)
C) \(400\)
D) \(200\)
E) \(50\)
Solution:
In \(\dfrac{dP}{dt} = kP\left(1 - \dfrac{P}{L}\right)\), the carrying capacity is \(L\).
Here \(L = 200\) (the value making \(\tfrac{dP}{dt} = 0\) for \(P > 0\)).
\(= \boxed{200}\)
Problem 30
Find the radius of convergence of \(\displaystyle\sum_{n=0}^{\infty} \dfrac{x^n}{2^n}\).
A) \(1\)
B) \(0\)
C) \(\dfrac{1}{2}\)
D) \(4\)
E) \(2\)
Solution:
This is geometric in \(\dfrac{x}{2}\): it converges when \(\left|\dfrac{x}{2}\right| < 1\).
\(|x| < 2\).
Radius of convergence \(= \boxed{2}\)