Final Exam Review

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In this section we prepare for the final exam.

1 Limits

(problem 1) $\lim _{x \to 2} \frac {x^2 - 4}{x^2 - 5x + 6} = \answer {-4}.$

$\lim _{x \to 3} \frac {x^2 -x -6}{x^2 - 9} = \answer {5/6}.$

(problem 2) $\lim _{x \to 4} \frac {\sqrt x - 2}{x^2 - 5x + 4} = \answer {1/12}.$

$\lim _{x \to 9} \frac {\sqrt x - 3}{x^2 - 8x - 9} = \answer {1/60}.$

(problem 3) $\lim _{x \to 2} \frac {\frac {1}{x} - \frac 12 }{x- 2} = \answer {-1/4}.$

$\lim _{x \to {-3}} \frac {\frac {1}{x} + \frac 13 }{x+ 3} = \answer {-1/9}.$

(problem 4) $\lim _{x \to \infty } \frac {x^2 - 5x + 4}{2x^2 + 4x + 7} = \answer {1/2}.$

$\lim _{x \to -\infty } \frac {3x^2 + x - 2}{x^3 -3x + 1} = \answer {0}.$

(problem 5) $\lim _{x \to 4^+} \frac {2x}{x-4} = \answer {\infty }.$

$\lim _{x \to 2^-} \frac {x+1}{x^2 - 4} = \answer {-\infty }.$

(problem 6) $\lim _{x \to 0} \frac {\sin (3x)}{x} = \answer {3}.$

$\lim _{x \to 0} \frac {e^{2x} -1}{x} = \answer {2}.$

(problem 7) $\lim _{x \to \infty } \frac {\ln (x)}{x} = \answer {0}.$

$\lim _{x \to \infty } \frac {e^x}{x^2} = \answer {\infty }.$

2 Derivatives

(problem 8) $\frac {d}{dx}\left (5x^4 - 7x^2 + 3x -3 \right ) = \answer {20x^3 - 14x + 3}.$

(problem 9) $\frac {d}{dx}\left (x^3 + \frac {1}{x^3} + \sqrt [3]x \right ) = \answer {3x^2 -3x^{-4} +(1/3)x^{-2/3}}.$

(problem 10) $\frac {d}{dx}\big [\tan (x) + \cot (x) \big ] = \answer {\sec ^2(x) - \csc ^2(x)}.$

$\frac {d}{dx}\big [\sec (x) + \csc (x) \big ] = \answer {\sec (x)\tan (x) - \csc (x)\cot (x)}.$

(problem 11) $\frac {d}{dx}\left (2^x + e^x \right ) = \answer {2^x \ln (2) + e^x}.$

(problem 12) $\frac {d}{dx}\big [\tan ^{-1}(x) + \sin ^{-1}(x) \big ] = \answer {\frac {1}{1+x^2} + \frac {1}{(1-x^2)^{1/2}}}.$

(problem 13) $\frac {d}{dx}\left (x^2 \sin (x) + \sin (x^2) \right ) = \answer {x^2\cos (x) + 2x\sin (x) + 2x\cos (x^2)}.$

(problem 14) $\frac {d}{dx}(x^3 + 2x^2 - 5x - 4 )^6 = \answer {6(x^3 + 2x^2 - 5x - 4 )^5(3x^2 + 4x -5)}.$

(problem 15) $\frac {d}{dx}\left (x^3 \ln (x) \right ) = \answer {x^2 + 3x^2\ln (x)}.$

(problem 16) $\frac {d}{dx}\left (e^{3x} \tan (2x)\right ) = \answer {3e^{3x} \tan (2x) + 2e^{3x}\sec ^2(2x)}.$

(problem 17) $\frac {d}{dx}\left (\frac {2x+5}{3x - 4}\right ) = \answer {\frac {-23}{(3x-4)^2}}.$

$\frac {d}{dx}\left (\frac {x}{x^2 + 1}\right ) = \answer {\frac {1 - x^2}{(x^2 + 1)^2}}.$

(problem 18) Find the equation of the tangent line to the graph of $y=\sqrt {x}$ at $x = 16$ , and use it to approximate $\sqrt {18}$ .
The equation is $y = \answer {x/8 + 2}$ and $\sqrt {18} \approx \answer {4.25}$ .

(problem 19) Find the equation of the tangent line to the graph of $y=1/x$ at $x = 3$ .
The equation is $y = \answer {-x/9 +2/3}$ .

3 Integrals

(problem 20) $\int \left (x^2 - 5x + 3 \right ) \ dx = \answer {x^3/3 - 5x^2 /2 + 3x} + C.$

(problem 21) $\int \left (x^3 + \frac {1}{x^3} + \sqrt [3]x \right ) \ dx = \answer {x^4/4 +x^{-2}/{-2} + (3/4)x^{4/3}} + C.$

(problem 22) $\int \big [2\sin (x) - 3\cos (x) \big ] \ dx = \answer {-2\cos (x) -3\sin (x)} + C.$

(problem 23) $\int \big [4\sec ^2(x) + 5 \csc ^2(x) \big ] \ dx = \answer {4\tan (x) - 5 \cot (x)} + C.$

(problem 24) $\int _0^2 \left (x(x+1) + 1 \right ) \ dx = \answer {20/3}.$

(problem 25) $\int _0^\pi \sin (2x) \ dx = \answer {0}.$

(problem 26) Find the area under the graph of $y = \sqrt x$ from $x=0$ to $x=16$ .
The area is $\answer {128/3}$ .

(problem 27) Find the area under the graph of $y = 1/x$ from $x=1$ to $x=3$ .
The area is $\answer {\ln (3)}$ .

4 FTC, Part II

(problem 28) $\frac {d}{dx} \int _0^x \sin (2t^2) \ dt = \answer {\sin (2x^2)}.$

(problem 29) $\frac {d}{dx} \int _x^0 \sin (2t^2) \ dt = \answer {-\sin (2x^2)}.$

(problem 30) $\frac {d}{dx} \int _0^{3x} \sin (2t^2) \ dt = \answer {3\sin (18x^2)}.$

(problem 31) $\frac {d}{dx} \int _0^{x^3} \sin (2t^2) \ dt = \answer {3x^2\sin (2x^6)}.$

5 Implicit Differentiation

(problem 32) Find $\dfrac {dy}{dx}$ if $x^3 + y^3 = 6xy$ . $\frac {dy}{dx} = \answer {\frac {2y - x^2}{y^2 - 2x}}.$

(problem 33) Find $\dfrac {dy}{dx}$ if $y^4 = x^4 - xy$ . $\frac {dy}{dx} = \answer {\frac {4x^3 - y}{4y^3 +x}}.$

6 Related Rates

(problem 34) The radius of a circle is increasing at a rate of $2$ cm/min. How fast is the area increasing when the radius is $10$ cm? $\text {The area is increasing at a rate of} \ \answer {40\pi } \ \text {cm$^2$/min}.$

(problem 35) The radius of a sphere is increasing at a rate of $2$ cm/min. How fast is the volume increasing when the radius is $10$ cm? $(V= \frac 43 \pi r^3)$ $\text {The volume is increasing at a rate of} \ \answer {800\pi } \text {cm$^3$/min}.$

(problem 36) The short leg of a right triangle is increasing at a rate of $2$ cm/min and the long leg is increasing at a rate of $3$ cm/min. How fast is the hypotenuse increasing when the short leg is $12$ cm and the long leg is $16$ cm? $\text {The hypotenuse is increasing at a rate of} \ \answer {3.6} \text {cm/min}.$

7 Optimization

(problem 37) A farmer has 1200 feet of fence with which to fence in a rectangular pasture. Moreover, he would like to partition the pasture into two plots with fence parallel to the bottom side (horizontal). What dimensions will maximize the total area of his enclosed pastures?
The optimal length (horizontal) is $\answer {200}$ feet and
the optimal width (vertical) is $\answer {300}$ feet.

(problem 38) A fifth grader has $1200$ cm $^2$ of cardboard with which to construct a box with a square base and an open top. What is the maximum possible volume for his box?
The maximum possible volume is $\answer {4000} \ \text {cm}^3$ .

8 Max/Min, Inc/Dec and Concavity

(problem 39) Find the absolute maximum and the absolute minimum of the function $f(x) = x^3 - 3x+ 5$ on the interval $[0, 2]$ .
The absolute maximum is $\answer {7}$ occurring at $x = \answer {2}$ .
The absolute minimum is $\answer {3}$ occurring at $x = \answer {1}$ .

(problem 40) Find the absolute maximum and the absolute minimum of the function $f(x) = \sin (x) + \cos (x)$ on the interval $[0, \pi ]$ .
The absolute maximum is $\answer {\sqrt 2}$ occurring at $x = \answer {\pi /4}$ .
The absolute minimum is $\answer {-1}$ occurring at $x = \answer {\pi }$ .

(problem 41) The function $f(x) = x^3 - 27x$ is increasing on the interval(s):

$(-\infty , -3)$ and $(3, \infty )$ $(-\infty , -3)$ only $(3, \infty )$ only $(-3, 3)$

(problem 42) The function $f(x) = x^3 - 27x$ is concave down on the interval(s):

$(-\infty , 0)$ $(-\infty , 6)$ $(0, \infty )$ $(0, 6)$

9 Rectilinear Motion

(problem 43) A projectile is launched vertically. Its height (in feet) after $t$ seconds is given by $s(t) = 12 + 64t -16t^2$ .

The maximum height of the projectile is $\answer {76} \ \text {feet}$ .
The velocity of the object when it is $60$ feet up and falling is $\answer {-32} \ \text {ft/sec}$ .

(problem 44) A projectile is launched vertically. Its height (in feet) after $t$ seconds is given by $s(t) = 8 + 48t -16t^2$ .

The maximum height of the projectile is $\answer {44} \ \text {feet}$ .
The velocity of the object when it is $40$ feet up and rising is $\answer {16} \ \text {ft/sec}$ .

10 Mean Value Theorem

(problem 45) Find the value of ‘ $c$ ’ from the Mean Value Theorem for the function $f(x) = 1/x^2$ on the interval $[1,2]$ $c=\answer {(8/3)^{1/3}}.$

(problem 46) Show that there is no value of ‘ $c$ ’ from the Mean Value Theorem for the function $f(x) = 1/x^2$ on the interval $[-1, 1]$ . Why does this NOT contradict the statement of the theorem?

11 Area

(problem 47) Find the exact area under the graph of $y = e^{2x}$ from $x = 0$ to $x = 1$ .

Area under the curve $= \int _a^b f(x) \ dx$

$\text {area under the curve} = \answer {(e^2 -1)/2}.$

(problem 48) Find the exact area above the graph of $y = 1 - \sqrt x$ from $x = 1$ to $x = 4$ .

Area above the curve $= -\int _a^b f(x) \ dx$

$\int _1^4 1 - \sqrt x \ dx = \answer {-5/3},$ $\text {area above the curve} = \answer {5/3}.$

2025-04-30 03:38:04