1. The graph of a function f is given above. Answer the question:
a. Find the value(s) of x where f is not differentiable.
b. Find the value(s) of x where f is not differentiable but is continuous.
c. Find the value(s) of x where f is not continuous, but has a limit.
d. Find the value(s) of x where f does not have a limit.
2. The graph of a function f is given above. Answer the question:
a. Find the value(s) of x where f is not differentiable.
b. Find the value(s) of x where f is not differentiable but is continuous.
c. Find the value(s) of x where f is not continuous, but has a limit.
d. Find the value(s) of x where f does not have a limit.
3. Find f 0 (x), or
dy
, as appropriate:
dx
a. f (x) = 3
b. f (x) = 4x + 1
c. f (x) = −2x2 − 3x + 1
3x3
4
√
e. f (x) = x − 1
√
√
√
f. f (x) = − x5 + 3 x3 + 7 x
d. f (x) =
g. f (x) = ex
h. f (x) = e
i. f (x) = 3xex
√
x+1
j. y =
3x − 4
k. y = (4x + 3)3
l. y = (x2 − 3x − 2)5
p
m. f (x) = 4x3 − 2x2 + 4x + 5
n. f (x) = e3x−1
o. y = ex
2
p. y = ln x
q. y = ln(x2 + 4x − 1)
1
x
s. y = ln(ln x)
r. y = ln
1
ln x
u. y = sin x
t. y =
v. f (x) = sin
3π
4
w. f (x) = 3x sin x + sin x cos x
x. f (x) = 3x2 ex + 4x tan x
y. f (x) = sin x cos xex
z. y = 3x2 sin x cos xex
2
aa. y = e4 sin(x
−1)
bb. y =
3ex − sin x
cos x − 4ex
cc. y =
9x2 ex + 4x cos x
√
2 sin x + xex
dd. y =
ex sin x − x cos x
3x cos x + 4xex
ee. f (x) = sin(3x2 + 1)
ff. f (x) = cos (sin (4x − 2))
gg. y = sin(e3x )
hh. f (x) = sin(4x)
ii. y = sin3 x
jj. y = cos4 (x2 + 1)
kk. y = ecos
2
x−1
ll. f (x) = tan3 e2x−1
mm. f (x) = cos(2x3 − 4)ecos x
nn. f (x) = x tan(x2 + 1)e3x
2
oo. f (x) = x2 esin x + ex sin(cos(3x − 1))
2
e + 3xex −2
e − 4 sin x cos2 x
pp. y =
qq. y = tan5 sec3 x2 + 3
rr. y = arccos x
ss. y = sin5 e3π+5 + 4e3 x2
tt. y = arcsin(x2 − 5x + 1)
uu. y = arctan2 (ln 3x)
ln(3x)
sin(ln x2 )
vv. y =
ww. y =
3e4x sin−1 (3x2 + 1)
ln x sin 5x + 3x2 cos2 x
4. Find the equation of the line tangent to the given function f at the given point x = a:
a. f (x) = 2, a = 3
b. f (x) = 5x − 1, a = −2
c. f (x) = x2 − 2x + 4, a = 1
d. f (x) = ln x, a = 3
e. f (x) = e3x , a = −2
f. f (x) = xex − ln 4x, a = 1
π
g. f (x) = sin x, a =
6
5.
a. Define a function that has a left and right handed limit at a point a, but f does not have a limit at a.
b. Define a function that has a limit at a point b but is undefined at b.
c. Define a function that has a limit at point c, is defined at c, but is discontinuous at c.
d. Define a function that is continuous at point d but is not differentiable at d.
e. Define a function that is first differentiable at point p but is not second differentiable at point p.
f. Draw the graph of a function which has all of the properties in a, b, c, d, e, above.
6. Think About:
a. Given the graph of the original function f (x), how can you draw the graph of its derivative, f 0 (x)?
b. Given the graph of the derivative of a function, f 0 (x), how can you draw the graph of the original function,
f (x)?
7. Find
dy
by implicite differentiation:
dx
a. y sin x = x cos y
b. x2 y + y 2 x = 5
c. 2y − 4x sin x = ey
x
d. ln(xy) + = 1
y
e. exy + ln(x + y) = 2xy
8. Find f 00 (x) and f 000 (x):
a. f (x) = 4
b. f (x) = 3x2
c. f (x) = ex
d. f (x) = ln x
x2 + 1
x−1
√
f. f (x) = x
e. f (x) =
g. f (x) = x5/4
h. f (x) = arccos x
i. f (x) = sin x
j. f (x) = esin x
9. Find the linearization of the given function at the given point a:
a. f (x) = 2x − 3, a = 2
b. f (x) = 2x − 3, a = 4
c. f (x) = 2x − 3, a = 10
d. f (x) = 4x2 − x + 3, a = −1
e. f (x) = sin x, a = 0
π
f. f (x) = cos x, a =
4
x
g. f (x) = e , a = 0
h. f (x) = ex , a = 1
i. f (x) = ln x, a = 1
j. f (x) = ln x, a = 2
k. f (x) = arctan x, a = 0