Math 157 Workshop 3, 12/13/2016
1. Find the following limits:
√
2x | x − 1|
• lim
−
x →1 2x2 + x − 3
df
2. Find f 0 ( x ) =
if
dx
• f ( x ) = sin3 (12x2 + 5)
√
3
• lim
x →8
• f (x) =
x−2
x−8
p
4
1+
√
x − tan x
x →0 x − sin x
• lim
x
• f ( x ) = tan
cos x x
3. Use implicit differentiation to find the equation of the tangent line to the curve
x2 + y2 + xy − 3 = 0
at the point (1, 1).
4. For what value of the constants A, B is the function
 2
if x ≤ 1
− x + A
f (x) = B

if x > 1
x
differentiable everywhere?
5. Show that the equation x5 + x = 4 has a unique solution.
6. Let f be a twice differentiable function. Suppose there are numbers a < b < c
such that f ( a) = f (b) = f (c) = 0. Show that there is an x for which f 00 ( x ) = 0.
7. A reservoir has the shape of an upside down circular cone with height 10 ft
and base radius 4 ft. Water is being pumped out at the constant rate of 5 ft3 /min.
How fast is the water level falling when the depth of the water is 5 ft?
8. Find the trapezoid of largest area that can be inscribed in a semicircle of radius
a, with one base lying along the diameter.
9. Sketch a possible graph for a function f with the following properties:
•
•
•
•
The domain of f is all x 6= 1; the line x = 1 is a vertical asymptote of f .
limx→∞ f ( x ) = −1 and limx→−∞ f ( x ) = 1.
f 0 (0) = 0, f 0 ( x ) > 0 when x < 0, f 0 ( x ) < 0 when 0 < x < 1 or x > 1.
f 00 (−1) = 0, f 00 ( x ) > 0 when x < −1 or x > 1; f 00 ( x ) < 0 when −1 < x < 1.
2
10. Sketch the graph of the function f ( x ) = 4x/( x2 + 4) on the real line. Get
as much information as you can about the increase/decrease, critical points,
convexity, and inflection points of f .
11. Determine the number of solutions of the equation
3x4 − 4x3 − 12x2 + 2 = a.
Your answer will of course vary depending on what the constant a is.
12. Verify that the function f ( x ) = 2x + cos x is one-to-one, so its inverse f −1
exists. Find the derivatives ( f −1 )0 (1) and ( f −1 )0 (π ).
Hints.
1. For the last limit, apply L’Hôpital twice.
3. Take the derivative of each side of the equation with respect to x, keeping in mind that y is a
function of x. Then solve the resulting equation for y0 = dy/dx. Compute y0 at the point x = y = 1
to find the slope of the tangent line.
4. Only differentiability at x = 1 is in question. First write the condition of f being continuous at
x = 1. Then write the condition of f having the same slope when x = 1 is approached from left
and right (to avoid a sharp corner there). The two conditions will determine A and B.
5. Let f ( x ) = x5 + x − 4. First apply IVT to show that f ( x ) = 0 has at least one solution. To prove
uniqueness of this solution, verify that f is increasing.
6. Use Rolle’s Theorem.
7. Let V = V (t) and h = h(t) denote the volume and depth of water in the reservoir at time t. Find
a relation between V and h at any time, differentiate it with respect to t to find a relation between
V 0 = dV/dt and h0 = dh/dt. Since you know V 0 , you can compute h0 . You will need the formula
V = (1/3)πr2 h for the volume of a cone of base radius r and height h.
8. Find h in terms of the length x shown in the
picture. Then use it to express the area A of the
trapezoid as a function of x only. Use calculus to
find the maximum value of A over the interval
0 ≤ x ≤ a.
h
x
a
9. It is best to put the given information in a table and then proceed as usual.
11. Determine the intervals of increase/decrease of the function f ( x ) = 3x4 − 4x3 − 12x2 + 2 to get
an idea of what the graph of f looks like.
12. Show that f is strictly increasing. To find the required derivatives using the formula for ( f −1 )0
you will need the values of f −1 (1) and f −1 (π ), which are easy to find by try and error.