MATH 140 SPRING 2023 PRACTICE PROBLEMS FOR FINAL EXAM
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MATH 140 SPRING 2023 PRACTICE PROBLEMS FOR FINAL EXAM
Please keep in mind that this list is only a guideline. You should not assume that every problem on the
actual final exam will be a slight modification of some problem listed below.
The use of calculators of any kind will not be permitted on the final exam, nor will the use of any kind of
formula sheet. For this reason it is in your best interest to work through all of the problems without the aid
of a calculator or an app, and without looking up formulas.
When working on problems that involve limits, keep in mind that on the final exam, zero credit will be
given for any solutions which use L’Hospital’s Rule, regardless of what your instructor may have allowed
on your earlier assessments.
北h(1.)otes the point (x, 北1 ), find the simplified
(2) In each case, evaluate the limit. If the limit is not a number, write −∞ , ∞ , or DNE, and justify your
answer.
(a) 北2 x2 1(2)0x(−) (b)u(l) (c)t(l) ( − )
(d) r (e)q(l) q3 sin ( )
(3) The graph of a function f is given below. In each case, evaluate the given limit or explain why it
does not exist.
(a) 北+ f(x)
(b) 北 + f(x)
(c) l北 f(x)
(d) 北 f(x)
(e) 北 ∞ f(x)
(4) Find all horizontal and vertical asymptotes of the function
x2 + 1
3x2 − 7x + 4
All lines must be described by equations.
(5) Consider the function
'5x − 8, x < 2
t2 + 3 , t ≤ 3
g(t) =〈
( 8 , t > 3
(7) Use the Intermediate Value Theorem to show that the equation
4e北 = 3x + 5
has at least one solution in the interval (0, 1). All the hypotheses of the Intermediate Value Theorem
must be checked.
(8) In each case, evaluate the limit. If the limit is not a number, write −∞ , ∞ , or DNE, and justify your answer.
(a) u ∞ (b) 北 x(−) (c) t (d) t
s(t) = 2t3 − t + 1
type of problem appears on the final, no credit at all will be given for using anything other than the
definition of the derivative.)
(11) If f and g are differentiable functions such that f(3) = 5,f′ (3) = −8,g(3) = −2 and g ′ (3) = 7,
(a) (2f − 5g)′ (3) (b) (fg)′ (3) (c) ( ) ′ (3) (d) (f(f) g(g) )′ (3)
(12) In each case, find the derivative of the given function in simplest form.
(a) g(t) = t5 + t(6)t32 − 1 (b) f(x) = − +^e
(17) A sugar cube slowly dissolves while keeping its cubic shape. The volume is decreasing by 0.03 cm3
per second. How quickly is the height of the cube decreasing when the cube has volume 27 cm3 ?
Your answer should be in the correct units.
(18) The radius of a spherical ball is increasing a rate of 2 cm / min. At what rate is the surface area of the ball increasing when the radius is 10 cm? (Recall that the surface area of a sphere of radius r is 4πr2 .) Your answer should be in the correct units.
(19) Use linear approximation to estimate ^ . Without using a calculator or a graphing utility, deter-
mine whether your linear approximation of this quantity is an overestimate or an underestimate.
(Hint: consider concavity.)
(20) In each case, determine all the critical numbers of the given function, and determine whether a local
maximum, a local minimum, or neither is attained at the critical numbers.
(a) f(x) = 6x5 − 30x4 + 40x3 + 5 (b) g(t) = (t2 + 1)et (c) h(r) =
(21) Find the absolute maximum and absolute minimum values of the function f(x) = x + on the
(22) Does there exist a function g continuous on [2, 8] and differentiable on (2, 8) such that g(2) = 7,g(8) =
5 and g ′ (x) ≥ −0 .33 for all x in (2, 8)? Justify your answer.
(23) If h(x) = , find the average rate of change of h from x = 1 to x = 5, and find all c in (1, 5)
(iv) intervals of concavity up and/or concavity down
Use these to sketch the graph of f.
(25) Find the maximum possible value of the product xy if x + 2y = 5.
(26) If a cylindrical container with open top is made from 100 cm2 of material, find the largest possible
volume of the container. Your answer should be in the correct units.
(27) In each case, find the most general antiderivative of the given function.
(a) f(x) = 2x5 x2(7x) + 1 (b) g(r) = 6er − 5sin r + 2 (c) h(w) = 3secw tanw
(28) The following quantity represents the area under the graph y = g(x) from x = a to x = b. Find
g(x),a and b.
nl · 41 − ( + )4
(29) If the graph of f is given below, evaluate each of the following definite integrals by interpreting
them in terms of net area.
(a) \1 7 f(x) dx (b) \9 3 f(x)dx (c) \0 5 (2f(x) − 9) dx (30) If F(x) = \x5(2) dt, find F′ (x) in simplest form.
(b) \ dr
2sinθ
1 − cosθ
(c) \ dw
(c) \0 3 dt
2023-05-19