Tutorial 8
Instructions: No Calculators; Show all workings
1. For what values of c does ln x = cx2 have a solution. Assume that c > 0.
2. For which values of a and b is (1, 6) a point of inflection of the curve y = x3 + ax2 + bx + 1?
3. Sketch the graph of f (x) = 3x4 − 8x3 + 10, after answering the following questions
(a) Where is the graph increasing and decreasing?
(b) Where is the graph concave up and concave down?
(c) Where are the local minima and local maxima?
(d) Where are the inflection points?
(e) What happens to f (x) as x → ∞ and x → −∞?
4. Let
6x3 − 1
f (x) =
−2x2 + 18
(a) Find the horizontal and vertical asymptote of f (x).
(b) Show that y = −3x is a slant asymptote of f (x).
(c) Find the x-intercepts and y-intercepts of f (x).
(d) Sketch the graph of f (x).
5. Let
x2 − 1
f (x) =
x2 − 4
(a) Find the x and y-intercepts of f .
(b) Determine whether f is even, odd.
(c) Determine the vertical and horizontal asymptotes of f .
(d) Find the limx→−2+ f (x) and limx→2− f (x).
(e) Find the limx→−2− f (x) and limx→2+ f (x).
(f) Determine the concavity of f and find all inflection points.
(g) Using your answers to (a) − (f ), sketch the graph of f .
(h) Use your graph to determine the intervals on which f is increasing or decreasing.
(i) Find and classify all local extrema of f using your graph.
x−3
6. Consider the function f (x) = √ .
x2 − 9
(a) Find the domain of f .
(b) Find the coordinate of all x- and y-intercepts, if any.
(c) Find all horizontal and vertical asymptotes, if any.
(d) Find all critical numbers, if any.
(e) Find all intervals on which f is increasing and those on which f is decreasing.
(f) Find the (x, y) coordinates of all maximum and minimum points.
(g) Find all intervals on which f is concave up and those on which f is concave down.
(h) Find the (x, y) coordinates of all inflection points
(i) Sketch the graph of f (x) using all the above information. All relevant points must be labeled.
7. Sketch the following graphs
x3 − 2x 2
(a) f (x) = , (b) f (x) = (5 − 2x)x 3 (c) f (x) = x2 e−x
3x2 − 9
8. Let f (x) be a function whose domain is R. It is known that:
f (x) > 0 for all real x;
f (x) is differentiable at x = 0;
f (x + y) = f (x)f (y) for any real x and y.
Show that f (x) is differentiable in R.
9. (a) State the conclusion of Rolle’s Theorem.
(b) Prove that the equation x3 − 15x + d = 0
has at most one root in the interval [−2, 2].
(c) Determine whether the hypotheses of Rolle’s theorem hold for the function f (x) on the given
interval, and, if they do, verify the conclusion of the theorem.
i. f (x) = x2 − 2x − 3 on [−1, 3]
ii. f (x) = x3 − x on [0, 1]
x+3
iii. f (x) = on [1, 3]
x−4
p
iv. f (x) = 25 − x2 on [−3, 4]
v. (
x3 −2x2 −5x+6
x−1 if x 6= 1, and x ∈ [−2, 3]
f (x) =
−6 if x = 1,
vi. (
x2 if 0 ≤ x ≤ 1,
f (x) =
2 − x if 1 ≤ x ≤ 2,
10. (a) State the conclusion of the Mean Value Theorem.
(b) Prove that, if f 0 (x) > 0 for all x in the open interval (a, b), then f (x) is an increasing function
on (a, b).
(c) Let f (x) = 2 − |2x − 1|.
Show that there is no value of c such that f (3) − f (0) = f 0 (c)(3 − 0).
(d) Why does this not contradict the Mean Value Theorem?
!
3x
(e) Verify that the function g(x) =
x+7
satisfies the hypothesis of the Mean Value Theorem on the interval [−1, 2]. Then find all
numbers c that satisfy the conclusion of the MVT. Leave your final answer exact.
(f) Suppose g(x) is a differentiable function and g(2) = 1.
If −1 ≤ g 0 (x) ≤ 2 on [2, 5], how large or small g(x) can be?
(g) Show that y = 3x3 + 2x + 12 has a unique root
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(h) Prove that f (x) = − 2x + sin x has exactly one positive root.
(x + 1)2
πx
(i) Explain carefully why the equation 4x − 2 cos = 0 has exactly one real root.
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(j) Use the mean value theorem to prove that tan x > x for 0 < x < π2 .
(k) State Fermat’s Theorem, clearly identifying any hypothesis and the conclusion.
(l) State the Extreme Value Theorem, clearly identifying any hypothesis and the conclusion.
(m) Suppose g(x) is a function that is differentiable for all x. Let h(x) be a new function defined
by h(x) = g(x) − g(2 − x). Prove that h0 (x) has a root in the interval (0, 2).
(n) Show that
1
1− < ln x < x − 1 for any x>1
x
(o) Show that
√ x
1+x<1+
2
for x > 0 and for −1 ≥ x < 0
(p) Show that
√ √ y−x
y− x< √
2 x
if 0 < x < y
(q) Show that the equation x + cos x + ex = 0 has exactly one real root.
(r) Show that the equation
x5 + 4x = 1
has exactly one solution.
(s) Prove that x3 + px + q = 0 has exactly one real root if p > 0.
(t) Show that the equation 3 tan x + x3 = 2 has exactly one solution in the interval [0, π4 ].
(u) Show that
| sin x − sin y| < |x − y|
for all real x and y
11. The function (
ex if x ≤ 1,
f (x) =
mx + b if x > 1,
is continuous and differentiable at x = 1. Find the values for the constants m and b
12. Let f be a function continuous on [0, 1] and differentiable on (0, 1).
Also let f (0) = f (1) = 0. Show that there exists a point c in (0, 1) such that f 0 (c) = f (c).
13. Textbook Problems
Exercise Set 4.8 [1, 4, 7, 8, 11, 12, 22, 25, 29, 33, 41, 42]
Chapter 4 Review Exercise Numbers [3-10, 24-26, 37-44, 54-56, 75-78]
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