Calculus I Final Exam Spring 2018-19

Final Examination Question Paper (Spring 2018-19)

• Course Code: MATH 199
• Course Title: CALCULUS I
• Date of Exam: 23-05-2019
• Time: 09:00-11:00
• Total Questions: 9
• Total Marks: 40
• Question Paper Details: The examination paper consists of five printed pages, including the cover page and the last page. If a page is missing, inform the proctor immediately.

Useful Formulae

• Average Rate of Change: f(x₂)-f(x₁))/(x₂-x₁)
• Derivative of a function: lim (h→0) [f(x+h)-f(x)]/h
• Derivative Formulas:
  • d(e^u)/dx = e^u du/dx
  • d(u^m)/dx = mu^m-1 du/dx
  • d(a^u)/dx = a^u ln(a) du/dx
  • d(log_au)/dx = (1/(u ln(a))) du/dx
  • d(sin u)/dx = cos u du/dx
  • d(cos u)/dx = -sin u du/dx

Examination Questions

• Question 1: Find the domain of f(x) = √(x-16).

• Question 2: Find the limits of the functions;

  • (a) lim x→-1 (3x+5)/2
  • (b) lim x→2 (x²-7x+10)
  • (c) lim x→0 (5x⁴+4-2)/x
  • (d) lim x→0 (x²-2x+2sin6x)/(2xcos x)
  • (e) lim x→∞ (3x²+x²+1) / (x⁵+1)

• Question 3: Given function f(x);

  • (a) Find lim x→3⁺ f(x) and lim x→3⁻ f(x)
  • (b) Find lim x→3 f(x)
  • (c) Find lim x→3 f(x)
  • (d) Is f(x) continuous at x = 3? Justify your answer.

• Question 4: Find the average rate of change of f(x)= x²+2x over the interval [-1, 1].

• Question 5:

  • (a) Find the derivative of f(x)= 2x²-3 using the definition at x=1.
  • (b) Find the equation of the tangent line to the curve at x=1.

• Question 6: Find the first derivative of the following functions:

  • (a) f(x) = 4(2x²+1)
  • (b) f(x) = sin(3x³ + 2x)
  • (c) f(x) = e^{(x² sin x)}
  • (d) f(x) = ln(x² + 3x² + 2x)

• Question 7: Find dy/dx of xy + y² = 1 using implicit differentiation.

• Question 8: Find dy/dx of y = (sin x)^x using logarithmic differentiation.

• Question 9: Given f(x)=x³+3x²-18x+2;

  • (a) Find the critical points of f(x).
  • (b) Find the intervals where f(x) is increasing or decreasing.