MATH 181 University of Illinois Calculus I Questions

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Name: ______________________________ Calculus 1 – SUMMER 2020 Directions: Answer any 16 of the 18 questions. Show all work. You may use calculators, but your answers must be supported by your written work. 1. Evaluate the limit. 2. A rectangle is enclosed by the x-axis, y-axis, and the line 𝑦 = − 𝑥 + 3. Determine the length and width that give the rectangle the largest possible area. 3. Find all values of k such that 𝑓(𝑥) would be continuous. Based on the value of k from above, will this function be differentiable at x =2? Why or why not? 4. Evaluate: (4𝑥 − 2 )𝑑𝑥 𝑥 5. 6. Use the given graph of 𝑓(𝑥) to evaluate the following limits. a) lim → 𝑓(𝑥) = b) lim → 𝑓(𝑥) = c) lim → 𝑓(𝑥) = Determine all relative minimum and relative maximum for 𝑓(𝑥) = 𝑥 + − 2. 7. Use Newton’s Method to find the third iteration (that is, x 3) for: Use a starting value of x1 = 2. 8. Determine the absolute minimum and absolute maximum for 𝑓(𝑥) = 2𝑥 − 12𝑥 on [0, 4]. 𝒙𝟑 − 𝟐𝒙𝟐 − 𝟏 = 𝟎 ∫ 𝑓(𝑡)𝑑𝑡 9. Given the graph of 𝑓(𝑡), find: 10. The position of a particle moving along a straight line is given by 𝑠(𝑡) = 𝑡 − 3𝑡 + 4. a) Determine the velocity function; b) Determine at what time(s) the particle’s velocity is zero. c) Determine the acceleration function; d) Determine at what time(s) the particle’s acceleration is zero. 𝑣(𝑡) 𝑎(𝑡) 11. 12. Given: 𝑓(𝑥) = 𝑥 + 𝑥 − 𝑥 a) Determine the x-values for all inflection points of the function. b) State the intervals where the function is concave UP and concave DOWN. Given, 𝑓(𝑥) = 𝑥 − 2𝑥 + 3 , find 𝑓 (𝑥) using the Definition of the Derivative. 13. Find all values c that satisfy the Mean Value Theorem for 𝑓(𝑥) = 𝑥 − 𝑥 + 4 on [–3, 3]. 14. Use implicit differentiation to find the slope of the normal line of 4𝑥 + 𝑦 = 8 at (1, 2). 15. Use logarithmic differentiation to 𝑓 (𝑥) for 𝑓(𝑥) = 𝑥 . 16. A spherical balloon that is filling at a rate of 20π cm3/sec. At what rate is the radius changing when the radius of the balloon is 3 centimeters? 𝑉 = 𝜋𝑟 17. Find the equation of the tangent line for 𝑓(𝑥) = √𝑒 18. Use L’Hospital’s Rule to evaluate the given limit. lim ( ) → + 3 when x = 0.
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