Mathematics 265 Introduction to Calculus I

Sample Final Exam

Sample Final Examination 1

Time: 3.5 hours
Passing grade: 55%
Total points: 124

1. Compute the derivatives of each of the functions given below. You may not need to simplify your answers. (20 points)
   1. ${\displaystyle y=\text{sin }x · \text{cos}\left(\text{sin }{x}^2\right)}$
   2. ${\displaystyle y={\left(\frac{1+x^3}{1-x^2}\right)}^{\frac{1}{3}}}$
   3. ${\displaystyle y=\sqrt{1+\sqrt{1+x}}}$
   4. ${\displaystyle y=\frac{\sqrt{x^2-1}}{x^2-2x-8}}$
   5. ${\displaystyle y={sec}^2\left(\frac{x+1}{x-2}\right)}$
2. Use the method of differentials to estimate each of the values below to four decimal places (show the process). (8 points)
   1. ${\displaystyle \text{cos }62^{\circ }}$
   2. ${\displaystyle \sqrt{16\text{.}4}}$
3. Use the definition of the derivative as a limit to compute the derivative of ${\displaystyle \text{cot }x\text{.}}$ (6 points)
4. Sketch the graph of each of the functions below. State all critical points, cusps, vertical asymptotes and points of inflection. (20 points)
   1. ${\displaystyle f\left(x\right)=\frac{x^2}{x^2-1}\text{.}}$
   2. ${\displaystyle f\left(x\right)=x+\text{sin }x\text{.}}$
5. Find all maxima and minima of the functions listed below on the indicated intervals. (11 points)
   1. ${\displaystyle f(x)=2x^{\frac{5}{3}}-5x^{\frac{4}{3}}}$ on ${\displaystyle \left[-1\text{, }20\right]\text{.}}$
   2. ${\displaystyle f\left(x\right)=x+\text{cos }x}$ on ${\displaystyle \left[-\pi \text{, }2\pi \right]\text{.}}$
6. Find the dimensions of the rectangle of area ${\displaystyle 220{\text{ cm}}^2}$ that has the smallest perimeter. What is the perimeter? (6 points)
7. Use Newton’s method to approximate the solution of the equation
   ${\displaystyle x^4+2x-5=0}$ in the interval [1, 2]. (5 points)
8. Compute the value of each of the integrals listed below. (20 points)
   1. ${\displaystyle \int \left(x^2-x\right)\sqrt{3x}dx}$
   2. $\displaystyle \int {\sin (2x)\cos x\,dx}$
   3. $\displaystyle \int_a^b {\left( x + \cos (2x) \right) dx}$
   4. ${\displaystyle {\displaystyle \int \left(x^2-4\right)}{}^{2}dx}$
   5. ${\displaystyle {\int }_2^{4}x\sqrt{x-1}}dx$
9. Use the properties of the integral to find an interval where the value of the integral
   
   ${\displaystyle {\displaystyle {\int }_0^{\pi /3}\sec x}dx}$
   
   is located. (5 points)
10. Find the area between the curves ${\displaystyle y=x}$ and ${\displaystyle y=2-x^2\text{.}}$ (8 points)
11. Use the Fundamental Theorem of Calculus to evaluate (4 points)
    
    ${\displaystyle \frac{d}{dx}\underset{2x}{\overset{x}{\int }}\text{sin}\left(t^2\right)dt\text{.}}$
12. Water flows from the bottom of a storage tank at a rate of $r(t)=180-6t$ liters per minute, where $0\le t\le 50.$ Find the amount of water that flows from the tank during the first 15 minutes. (3 points)
13. A chain lying on the ground is 10 m long and its mass is 80 kg. How much work is required to raise one end of the chain to a height of 6 m? (5 points)
14. The temperature of a metal rod, 6 m long, is 3 x (in degrees centigrade) at a distance x meters from one end of the rod. What is the average temperature of the rod? (3 points)