Written Homework 5

Math 1200

Instructions: Submit your work on Gradescope bySunday, September 22, 2023 at 11:59

pm. You must show all work.

1.The equation to the tangent line of a functiong(x) atx= 4 is given byy=−3x+ 10.

Findg(4) andg

′

(4).

2.C(t) represents the amount of caffeine (in mg) in the bloodstreamthours after a cup

of coffee has been consumed. Describe the physical meaning of the quantityC

′

(3).

Include units in your answer.

3.Find the equation of the tangent line to the functionh(x) =

1

x+ 3

at the point



1,

1

4



.

Use the limit definition of the derivative to compute any necessary derivatives.

1

4.Each limit below represents the slope of the tangent line to the graph of a function

y=f(x) at a point (a, f(a)). Determine the functionf(x) and the value ofa.

(a) lim

x→3

|2−x|−1

x−3

(b) lim

h→0

e

h

−1

h

5.

Shown below is a portion of the graph of the functionf(x), which has domain [−5,6].

−5−4−3−2−1123456

−5

−4

−3

−2

−1

1

2

3

4

5

y=f(x)

x

y

(a) Find all points wheref(x) is not continuous.

(b) Find all points wheref(x) is not differentiable.

2

6.Consider the below graph of a functionf(x).Note that the graph is linear on the

interval (3,5).

123456789

−2

−1

1

2

3

4

5

y=f(x)

x

y

(a) Compute lim

h→0

f(4 +h)−f(4)

h

.

(b) Order the following from least to greatest:

i. The slope of the tangent line atx= 1.5

ii. The slope of the secant line on the interval [5,7]

iii.f

′

(3)

iv. The average rate of change off(x) on the interval [1,4]

v. The instantaneous rate of change off(x) atx= 8.

3

7. Using the limit definition of the derivative, find a formula forf

′

(x) whenf(x) =

1

√

x

.

4