Final Exam Review MATH 1426 (Show Work-type problems)
1. Find the value(s) of 𝑐 guaranteed to exist by the Mean Value Theorem:
𝑓(𝑥) = 𝑥 2 − 4𝑥 on [−1,3]
𝑓(𝑥) = 𝑥 3 on [−2,2]
𝑓(𝑥) = √𝑥 on [1,4]
𝑓(𝑥) = 𝑒 𝑥 on [0, ln 2]
2. Approximate the value of the expression using linearization methods, namely:
(i) use 𝐿(𝑥); (ii) use differentials:
√3.9
√4.2
3
√8.1
3
√7.8
3. State the type of indeterminate form the limit describes, then evaluate the limit.
lim (5𝑥)3𝑥
𝑥→0
5
lim (𝑥)𝑥
𝑥→∞
3 𝑥
lim (1 + )
𝑥→∞
𝑥
2
lim+(4𝑥 − 11)𝑥−3
𝑥→3
4. Find the average value of the given function on the specified interval:
𝑓(𝑥) = 𝑥 2 + 5 on [−2,2]
𝑓(𝑥) = 𝑥 3 on [0,2]
𝜋
𝑓(𝑥) = sin 𝑥 on [−𝜋, ]
2
5. Evaluate the Definite Integral:
𝜋
∫ sin 𝑥 cos 𝑥 𝑑𝑥
0
5
∫(3𝑥 + 5)7 𝑑𝑥
4
0
2
∫ 7𝑥𝑒 3𝑥 𝑑𝑥
−3
4
∫
1
𝑥
√𝑥 + 1
𝑑𝑥
6. Write a right-Riemann Sum with 𝑛 subintervals of equal length that approximates the definite
integral:
3
∫(−3𝑥 + 15)𝑑𝑥
0
3
∫(−3𝑥 + 15)𝑑𝑥
1
5
∫ 𝑥 2 𝑑𝑥
1
2
∫(3𝑥 − 1)2 𝑑𝑥
0
7. Use the Definition of the Definite Integral to evaluate the integrals from #6 above. Check your
solution using the Fundamental Theorem of Calculus.
8. Find the position function 𝑠(𝑡) that satisfies the following conditions:
𝑠(0) = 1; 𝑣(0) = 12; 𝑎(𝑡) = 3
𝑠(1) = 10; 𝑣(1) = 4; 𝑎(𝑡) = −5
𝑠(−3) = 0; 𝑣(−2) = −5; 𝑎(𝑡) = −32
9. Optimization.
What two nonnegative real numbers with a sum of 66 have the largest possible product?
Find positive numbers 𝑥 and 𝑦 satisfying the equation 𝑥𝑦 = 4 such that the sum 2𝑥 + 𝑦 is as small
as possible.
A storage shed is to be built in the shape of a box with a square base. It is to have a volume of 729
cubic feet. The concrete for the base costs $5 per square foot, the material for the roof costs
$8 per square foot, and the material for the sides costs $6.50 per square foot. Find the dimensions
of the most economical shed.
Squares with sides of length 𝑥 are cut out of each corner of a rectangular piece of cardboard
measuring 21 ft by 12 ft. The resulting piece of cardboard is then folded into a box without a lid.
Find the volume of the largest box that can be formed in this way.