Math 1A Final Practice Exam (Fall 2024) PDF

MATH 1A FINAL (PRACTICE 1) PROFESSOR PAULIN INSTRUCTIONS Do not turn over until instructed to do so. Write your name and SID in the spaces provided on one side of every page of the exam. This exam consists of 10...

MATH 1A FINAL (PRACTICE 1) PROFESSOR PAULIN INSTRUCTIONS Do not turn over until instructed to do so. Write your name and SID in the spaces provided on one side of every page of the exam. This exam consists of 10 questions. You have 3 hours to complete this exam. This exam will be electronically scanned. Do not add or remove any pages from the exam. There is an extra blank page for scratch work on the back of the exam. It can also be used as extra space to write formal solutions as long as everything is clearly labeled. Calculators are not permitted. Show as much working as possible. Even if you don’t end up with the correct answer, you may still get partial credit. Answers without justification will be viewed with suspicion and will not receive credit. Name: Student ID: GSI Name: Math 1A Final (Practice 1) PLEASE TURN OVER Name: SID: 1. (30 points) Calculate the derivatives of the following functions (you do not need to use the limit definition). (a) 2x ln(x) f (x) = sin(x) Solution (b) f (x) = ln |xsec(x) + 1| Solution PLEASE TURN OVER Name: SID: 2. Calculate the following limits. (a) sin(x) lim− x→0 x4 Solution (b) √1 lim+ cos(x) x x→0 Solution PLEASE TURN OVER Name: SID: 3. (30 points) Calculate the following integrals (you do not need to use Riemann sum definition). (a) √ Z ( 3 x − 1)2 dx Solution (b) Z e4 1 dx e3 x ln(x) − 2x Solution PLEASE TURN OVER Name: SID: 4. (30 points) Determine the local extrema and inflection points of the following function. f (x) = 5x2/3 + x5/3 Solution PLEASE TURN OVER Name: SID: 5. (30 points) Determine the equation of the tangent line at x = 1 of the following curve. Z 3x y= cos(πt)dt + 2x x Solution PLEASE TURN OVER Name: SID: 6. (30 points) Calculate the total area of the region enclosed by the curves y = ex and y = e2−x between x = 0 and x = 2. Solution PLEASE TURN OVER Name: SID: 7. (30 points) A company needs to design an open topped box with a square base. The box must have volume 32 (the units are unimportant). Find the dimensions of the box which minimize the amount of materials. Solution PLEASE TURN OVER Name: SID: 8. (30 points) A train accelerates and decelerates at a constant rate of 4 meters per sec- ond per second. The train’s maximum speed is 60 meters per second. Determine the maximum distance the train can travel in 1 minute if it starts and finishes at rest. Solution PLEASE TURN OVER Name: SID: 9. (30 points) Calculate the following limit. n X i2 lim n→∞ i=1 n3 + i3 Solution PLEASE TURN OVER Name: SID: 10. (30 points) A solid S has a base given by the ellipse x2 y 2 + = 1. 4 9 Cross-sections parallel to the y-axis and perpendicular to the base are equilateral trian- gles. Calculate the volume of S. √ 3 2 You may use the fact that an equilateral triangle with side length ℓ has area 4 ℓ. Solution END OF EXAM