StruggleFest Worksheet
March 10, 2015
Name:
MATH 160 section 002
1.
2. Compute the derivative of the function f (x) = x3 + 2|x| at the point x = 0 from the
definition of derivative at a point.
dg for the function g(t) = t2 |t| from the definition.
3. Find
dt t=0
4. Suppose f and g are functions. The following table gives values of f (x), g(x), f 0 (x)
and g 0 (x) at some particular values of x.
x f (x) g(x)
0
1
1
1
3
-4
f 0 (x) g 0 (x)
5
1/3
-1/3 -8/3
Find the derivatives with respect to x of the following combinations at the given
value of x
(a) 5f (x) − g(x), x = 1
(b) f (x)g(x)3 , x = 0
(c)
f (x)
,
g(x)+1
x=1
(d) f (g(x)), x = 0
(e) g(f (x)), x = 0
(f) (x11 + f (x))−2 , x = 1
(g) f (x + g(x)), x = 0
p
5. For the functions h(x) = 2|x − 2| and p(x) = |x|, find h0 (2) and p0 (0) from the
definition (since that is your only option). If these are not differentiable, explain why
and how they are different.
Find the derivatives of the following functions
1
6. f (x) = 2x + eπ 5 − x3 − √
4
x
7. x(t) =
4−t
.
t+2
8. g(x) =
(x + 2)(x + 1)
.
(x − 2)(x − 1)
9. v(x) =
10. f (s) =
11. h(x) =
3
7
− 3
2 − x 4x
√
3
s(7s2 − πs).
3x − x10 + 7x − 1
√
3− x
√
√
√
12. y = (x5 + x4 − x3 − x2 + x + 1)( x − 3 x + 4 x)
13. Find x00 (t),
d2 v
, and f (3) (x), where x(t), f (x) and v(x) are from numbers 6, 7, and 9.
dx2
14. For the above function, draw its derivative.
15. The following graph is of a function, f (x), and its derivatives f 0 (x) and f 00 (x).
Label which graph is f (x), which graph is f 0 (x), and which graph is f 00 (x).
Now let g = f 0 ◦ f , let h = f ◦ f 00 , and let F = h ◦ f . Use the above graphs to
estimate the following:
(a) g 0 (0)
(b) g 0 (−1)
(c) h0 (−1.5)
(d) h0 (1)
(e) F 0 (−0.5)
(f) F 0 (1.5)
16. Consider the curve Tschirnhausen cubic, y 2 − 3x2 = x3 .
(a) What is
dy
?
dx
(b) What is the slope of this curve at (−2, 2)? What is the equation of the tangent
line at (−2, 2)?
(c) What is the slope of the curve at (−3, 0)?
dx
at (−3, 0). What is happening at (−3, 0)?
dy
(This question is written correctly)
(d) Follow up question to part c: find
dy (e) Is there a tangent line to the curve at (0, 0) (using
)? What about
dx (0,0)
dx ? What is happening here?
dy (0,0)
17. Find
dr
for r2 + θ2 = 12.
dθ
18. Consider the curve:
(a) Find
du
.
dv
(b) Find
dv
.
du
u2 v 2 + 3 cos(πu) = cot(uv) + u + 7v.
19. Find the absolute maximum and absolute minimum of the function p(θ) = 3(θ)2/3 on
the interval [−27, 8].
20. Find the absolute maximum and absolute minimum of f (x) =
[0, 4].
√
x−
1
on the interval
2
21. Find the absolute extrema for h(z) = |z 3 − 9z| (on the entire interval).
True or False?
22. f (x) = |x| has infinitely many tangent lines at x = 0.
23. If the derivative does not exist at a point, then a tangent line cannot exist at that
point.
24. f (x) defined on [0, 6], then it must attain both an absolute maximum and an
absolute minimum on [0, 6].
25. f (x) = |x + 5| is not differentiable at every x in its domain.
26. If g(x) is a function that is continuous on the union of intervals [2, 4] ∪ [5, 7], then
g(x) must attain an absolute maximum on [2, 4] ∪ [5, 7].
27. If h(x) has a cusp at x = 15, then h(x) has a vertical asymptote at x = 15.
28. Piecewise functions are not differentiable.
29. Given that f (1) = f (3) = 0 and f 0 (2) = 0, then f (x) must be continuous on the
interval [1, 3].
30. A curve that doesn’t pass the vertical line test at x = −1 cannot have tangent lines
for points where x = −1.
31. If g(x) is not continuous, then g(x) is not differentiable.
32. If lim
h→0
f (2 + h) − f (2)
= 4, then f (2) = lim f (x)
x→2
4