# Calculus Worksheet.

Name: ______________________________ _______ MATH 181 – Midterm Exam Directions: Answer all questions. Show all work. You may use calculators, but your answers must be supported by your written work. If appropriate work is not shown, you may receive reduced or no credit at all. 1. Use the graph to evaluate each of the following. 2. Use the definition of the derivative to find f ‘ x , if f x 2 x 2 1 . Evaluate each indicated limit, if it exists. Show all algebraic work, as needed. 3. 5. 4. x3 lim 3 x 3 x 27 . 6. Find the derivative of each function. Use proper notation. 7. 9. 8. 𝑔(𝑥) = √2 − 3𝑥 10. 𝑓(𝑥) = 𝑐𝑜𝑠 (4x) − csc(6𝑥) 11. If 𝑓(𝑥) = 𝑒𝑥 sin(𝑥), find 𝑓′′(𝑥). 12. Find 𝑑𝑦 , 𝑑𝑥 given: 𝑒 − 7𝑥 = 𝑥 tan(𝑦) 13. Find all 𝒙 values where its tangents are horizontal for: 14. Find the equation of the tangent line at (𝟐, 𝟏) for: 𝑔(𝑥) = 𝑥 − 𝑥 − 3𝑥 + 4 𝑓(𝑥) = 15. A construction worker pulls a 5-meter plank straight up the side of a building (vertically) at a rate of 0.15 meters per second. How fast is the end of the plank sliding along the ground when it is 3 meters from the building? Name: ______________________________ _______ MATH 181 – Midterm Exam Directions: Answer all questions. Show all work. You may use calculators, but your answers must be supported by your written work. If appropriate work is not shown, you may receive reduced or no credit at all. 1. Use the graph to evaluate each of the following. 2. Use the definition of the derivative to find f ‘ x , if f x 2 x 2 1 . Evaluate each indicated limit, if it exists. Show all algebraic work, as needed. 3. 5. 4. x3 lim 3 x 3 x 27 . 6. Find the derivative of each function. Use proper notation. 7. 9. 8. 𝑔(𝑥) = √2 − 3𝑥 10. 𝑓(𝑥) = 𝑐𝑜𝑠 (4x) − csc(6𝑥) 11. If 𝑓(𝑥) = 𝑒𝑥 sin(𝑥), find 𝑓′′(𝑥). 12. Find 𝑑𝑦 , 𝑑𝑥 given: 𝑒 − 7𝑥 = 𝑥 tan(𝑦) 13. Find all 𝒙 values where its tangents are horizontal for: 14. Find the equation of the tangent line at (𝟐, 𝟏) for: 𝑔(𝑥) = 𝑥 − 𝑥 − 3𝑥 + 4 𝑓(𝑥) = 15. A construction worker pulls a 5-meter plank straight up the side of a building (vertically) at a rate of 0.15 meters per second. How fast is the end of the plank sliding along the ground when it is 3 meters from the building? Name: ______________________________ _______ MATH 181 – Midterm Exam Directions: Answer all questions. Show all work. You may use calculators, but your answers must be supported by your written work. If appropriate work is not shown, you may receive reduced or no credit at all. 1. Use the graph to evaluate each of the following. 2. Use the definition of the derivative to find f ‘ x , if f x 2 x 2 1 . Evaluate each indicated limit, if it exists. Show all algebraic work, as needed. 3. 5. 4. x3 lim 3 x 3 x 27 . 6. Find the derivative of each function. Use proper notation. 7. 9. 8. 𝑔(𝑥) = √2 − 3𝑥 10. 𝑓(𝑥) = 𝑐𝑜𝑠 (4x) − csc(6𝑥) 11. If 𝑓(𝑥) = 𝑒𝑥 sin(𝑥), find 𝑓′′(𝑥). 12. Find 𝑑𝑦 , 𝑑𝑥 given: 𝑒 − 7𝑥 = 𝑥 tan(𝑦) 13. Find all 𝒙 values where its tangents are horizontal for: 14. Find the equation of the tangent line at (𝟐, 𝟏) for: 𝑔(𝑥) = 𝑥 − 𝑥 − 3𝑥 + 4 𝑓(𝑥) = 15. A construction worker pulls a 5-meter plank straight up the side of a building (vertically) at a rate of 0.15 meters per second. How fast is the end of the plank sliding along the ground when it is 3 meters from the building? ..18 T-Mobile 10%- 10:03 AM Not Secure — math.utep.edu 3 of 3 Example: A construction worker pulls a five-meter plank up the side of a building under construction by means of a rope tied to one end of the plank. Assume the opposite end of the plank follows a path perpendicular to the wall of the building and the worker pulls the rope at a rate of 0.15 meter per second. How fast is the end of the plank sliding along the ground when it is 2.5 meters from the wall of the building? 2 2 2 dpull = 0.15 m/s o =5 dt with dy = 0.15 dt dy 5 sm y want to find dx dy o xty 个 >> NS 2x dx + 2 y 3 2x dx = -24 di with x’ty =25 and x=2.5, ‘ = dx = – 2y do dt (2,5)²ty”=25 6.25 + y²=25 -y dy de di y = 18,75 or y=818.75 We need x,y, and -y dx = -V18,75 (0.15)= -0.26 mls . x de The negative indicates direction, 2x 되었 – Х ② Finally صبا can find dx= dt 2.5 Example: A man 6 feet tall walks at a rate of 5 feet per second away from a light that is 15 feet above the ground. When he is 10 feet from the base of the light, a) at what rate is the tip of his shadow moving? P TIT b) at what rate is the length of his shadow changing? 15 dy 5 5 ( a) y = 2 x with dx = 5ft/sec and x = lof şls filse) = zz . $ lse med b) length of shadow is en Cy»»> = date = 8.33-533.33 ) 5 dx 3 To و لے 3 9y-15x als 3-х so y = 15th dy dx TE х 156y-x) = by 154-15x=by y = 5x Elo 5lm ..1 T-Mobile 7:05 PM 48% Done 20200622132542math_181_mid… ☺ all algebraic work, as needed. 4. – (4 + h)2 – 16 lim h→0 h 11 ch