Exam 4 Study Guide
1. Calculate the derivative of f (x) = xx using log-differentiation.
Z
2. Integrate
cot(x) dx.
3. Calculate the derivatives the following:
2
(a) f (x) = ex + (ex )2
(b) g(x) = ln
√
5x3 x + 9
x2 + x
(c) F (x) = x · ln(x)
(d) G(x) = log3 (x)
(e) A(t) = 10rt .
4. Differentiate f (x) = sin−1 (x) using the fact that it is the inverse of sin(x) and simplify
to an algebraic expression.
5.
ex − e−x
2
(a) Find the derivative of tanh(x),
sinh(x) =
and
cosh =
ex + e−x
2
(b) and simplify to a (combination of) hyperbolic function(s).
ex − x
6. Calculate the limit lim+ √
using L’Hospital’s rule if necessary.
x→0
x
ex − 1
using L’Hospital’s rule if necessary.
x→0 cos2 (x)
7. Calculate the limit lim
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8. Calculate the work required to lift a 100 m steel cable hanging from the top of a building
up to the top. The cable has mass .4 kg/m, and you can round the force of gravity to
10 N/kg (Newtons per kilogram).
9. Let f be the function f (x) = 4x2 − x4 from x = 0 to x = 2, and consider the solid of
revolution formed by rotating the function about the y-axis.
5
f (x) = 4x2 − x4
4
3
4
2
2
1
x
−2 −1
−1
1
2
3
4
5
2
0
1
−1
−2
0
0
1
−1
2 −2
(a) Write the volume of the solid from x = 0 to x = 2 as an integral.
(b) Find the volume of this solid by evaluating the integral.
10. Calculate the average value of f (x) = x2 + 3x over the interval [−1, 2].
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