MATH 1151Q-001 Fall 2015 Worksheet 1

MATH 1151Q-001
Fall 2015
Worksheet 1
1. Determine if the following are true or false and give justification for either answer.
x2 + 6x − 7
limx→1 x2 + 6x − 7
=
limx→1 x2 + 5x − 6
x →1 x2 + 5x − 6
(b) If limx→5 f ( x ) = 2 and limx→5 g( x ) = 0, then limx→5 [ f ( x )/g( x )] does not exists.
(a) lim
(c) If f is continuous at a, then f is differentiable at a.
(d) If f (1) > 0 and f (3) < 0, then there exists a number c between 1 and 3 such that f (c) = 0.
(e) If f 0 (r ) exists, then limx→r f ( x ) = f (r ).
(f) The equation x10 − 10x2 + 5 = 0 has a root in the interval (0, 2).
(g) The derivative of a polynomial is a polynomial.
(h) If g( x ) = x5 , then limx→2
g( x )− g(2)
x −2
= 80.
2. Compute the following limits.
( x + b)7 + ( x + b)10
4( x + b )
x →−b
(b) Let
(a) lim
(
g( x ) =
x2 − 5x
ax3 − 7
if x ≤ −1
if x > −1
Find the number a so that limx→−1 g( x ) exists.
x+1
− 4x2 + 4x
4x3
√
(d) lim
x →∞ 2x3 + 9x6 + 15x4
4x3
√
(e) lim
x →−∞ 2x3 + 9x6 + 15x4
(c) lim
x →0 x 3
sin 5x
x →0 3x
(f) lim
(g) lim
x →0
(h)
sin 3x sin 5x
x2
1 − tan x
x →π/4 sin x − cos x
lim
3. Compute the derivative using the limit definition.
(a) f ( x ) = mx + b
(b) g( x ) =
√
9−x
4. Compute the derivative using the rules of differentiation and simplify where possible.
(a) f ( x ) = 32 x3 + πx2 + 7x + 1
x3 − 4x2 + x
x−2
ex
f (x) = 2
x −1
f (t) = 5t2 + sin t
2 cos x
y=
1 + sin x
x cos x
y=
1 + x3
(b) g( x ) =
(c)
(d)
(e)
(f)
√
(g) f ( x ) = (2 x − 1)(4x + 1)−1
√
(h) f ( x ) = e
x
(i) f ( x ) = (2x6 − 3x5 + 3)25
(j) g( x ) = 5x + sin3 x + sin x3
(k) f ( x ) = csc5 3x
(l) h( x ) = xe−10x
(m) f ( x ) = xe x cot x
(n) y = tan−1 ( x2 )
(o) f (t) = arcsin(1/t)
5. Compute dy/dx by implicit differentiation.
(a) x3 − xy2 + y3 = 1
(b) xey = x − y
(c) cos( xy) = 1 + siny
p
(d) xy = x2 + y2
6. Compute an equation of the tangent line to the curve at the given point.
(a) y = 4 sin2 x, (π/6, 1).
(b) y sin(2x ) = x cos(2y), (π/2, π/4).
7. Compute y” if x6 + y6 = 1.
8. Related Rates
(a) Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius
of the oil spill increases at a constant rate of 1 m/s, how fast is the area of the spill
increasing when the radius is 30m?
(b) The length of a rectangle is increasing at a rate of 8 cm/s and its width is increasing at a
rate of 3 cm/s. When the length is 20 cm and the width is 10 cm, how fast is the area of the
rectangle increasing?
(c) A cylindrical tank with radius 5 m is being filled with water at a rate of 3 m3 /min. How
fast is the height of the water increasing?
(d) The radius of a sphere is increasing at a rate of 4 mm/s. How fast is the volume
increasing when the diameter is 80 mm?
(e) A street light is mounted at the top of a 15-ft-tall pole. A man 6 ft tall walks away from
the pole with a speed of 5 ft/s along a straight path. How fast is the tip of his shadow
moving when he is 40 ft from the pole?
(f) At noon, ship A is 150 km west of ship B. Ship A is sailing east at 35 km/h and ship B is
sailing north at 25 km/h. How fast is the distance between the ships changing at 4:00 PM?
(g) A baseball diamond is a square with side 90 ft. A batter hits the ball and runs toward first
base with a speed of 24 ft/s. At what rate is his distance from second base decreasing
when he is half to first base?
(h) Gravel is being dumped from a conveyor belt at a rate of 30 ft3 /min, and its coarseness is
such that it forms a pile in the shape of a cone whose base diameter and height are always
equal. How fast is the height of the pile increasing when the pile is 10 ft high?
(g)
(h)