Ma 113 Functions And Inverse Functions, The Exponential Function And The Logarithm Worksheet Page 21
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41Worksheet # 16: Review for Exam II
1. (a) State the definition of the derivative of a function f (x) at a point a.
(b) Find a function f and a number a such that
f (x)
f (a)
ln(2x
1)
=
x
a
x
1
(c) Evaluate the following limit by using (a) and (b),
ln(2x
1)
lim
x
1
1
2. State the following rules with the hypotheses and conclusion.
(a) The product rule and quotient rule.
(b) The chain rule.
1
3
2
3. A particle is moving along a line so that at time t seconds, the particle is s(t) =
t
t
8t meters
3
to the right of the origin.
(a) Find the time interval(s) when the particle’s velocity is negative.
(b) Find the time(s) when the velocity is zero.
(c) Find the time interval(s) when the particle’s acceleration is positive.
(d) Find the time interval(s) when the particle is speeding up. Hint: What do we need to know about
velocity and acceleration in order to know that the derivative of the speed is positive?
4. Compute the first derivative of each of the following functions:
1
3
(a) f (x) = cos(4πx
) + sin(3x + 2)
(g) m(x) =
x +
4
x
4
2
(b) b(x) = x
cos(3x
)
sec(2 )
(c) y(θ) = e
e
(h) q(x) =
2
2
1 + x
(d) k(x) = ln(7x
+ sin(x) + 1)
1
2
(e) u(x) = (sin
(2x))
(i) n(x) = cos (tan(x))
2
8x
7x + 3
(f) h(x) =
(j) w(x) = arcsin(x) arccos(x)
cos(2x)
(4)
5. Let f (x) = cos(2x). Find the fourth derivative at x = 0, f
(0).
6. Let f be a one to one, differentiable function such that f (1) = 2, f (2) = 3, f (1) = 4 and f (2) = 5.
1
Find the derivative of the inverse function, (f
) (2).
x
7. The tangent line to f (x) at x = 3 is given by y = 2x
4. Find the tangent line to g(x) =
at
f (x)
x = 3. Put your answer in slope-intercept form.
3
2
2
8. Consider the curve xy
+ 12x
+ y
= 24. Assume this equation can be used to define y as a function
of x (i.e. y = y(x)) near (1, 2) with y(1) = 2. Find the equation of the tangent line to this curve at
(1, 2).
9. Each side of a square is increasing at a rate of 5 cm/s. At what rate is the area of the square increasing
2
when the area is 14 cm
?
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