Final Exam – Math 1071 – Business Math II – All Problems 2 points each – SHOW WORK
STUDY GUIDE
1. Evaluate these definite integrals.
a)
b)
2. Find the area enclosed by y = , y = , x= , and x =.
3. Find the area enclosed by y = and y = .
4. Find the coefficient of inequality of the following Lorentz curve:
f(x) = .
5. Integrate by parts:
a)
b)
6. Find the coefficient of inequality for the Lorentz curve y =
7. Evaluate each improper integral:
a)
b)
8. Solve each exponential and logarithmic equation for x.
a)
b)
𝑑𝑦
𝑑𝑦
𝑓(𝑥+ℎ)−𝑓(𝑥)
For problems 9 and 10 find 𝑑𝑥 using the limit definition 𝑑𝑥 = lim
.
ℎ
ℎ→0
9. f(x) =
10. f(x) =
𝑑𝑦
11. Now use the “shortcut” polynomial rules to find 𝑑𝑥 for
a) f(x) =
b) f(x) =
12. Find the equation of the tangent line to f(x) = at the point c =.
13. Use the product rule to find the derivatives of
a) y =
b) y =
14. Use the quotient rule to find the derivatives of
a) f(x) =
b) y =
15. Use the Chain Rule to find the derivatives of
a) f(x) =
b) y =
16. Find the derivatives of
a) f(x) =
b) f(x) =
17. Find all critical points, find all largest open intervals on which f(x) is increasing and
decreasing, find all relative maxima and minima. Draw a rough graph of f(x).
f(x) =
For problems 18 and 19 find all critical points, inflection points, and the largest open
intervals on which f(x) is increasing, decreasing, concave up, and concave down. Find all
relative maxs and mins. Graph
18. f(x) =
19. f(x) =
20. Find all absolute maxs/mins for f(x) = on [ , ].
21. Optimization problem
22. Optimization problem
23. Find the integrals
a)
b)
24. word problem using integration
25. Find the indefinite integrals by the substitution method.
a)
b)