Deck 10: An Introduction to Calculus: Limits, Derivatives, and Integrals

Question 1

(a) Explain why direct substitution cannot be used to find the limit
(b) Find the limit algebraically, if it exists.

Answer 1

(a) is not defined at x=-4 .
(b) 48

Question 2

What is the derivative of

Answer 2

8X

Question 3

Find the instantaneous rate of change of at x=3 .

Answer 3

30

Question 4

Determine the following limits, if they exist.
(a)
(b)
(c)
(d)
(e)
(f)

Question 5

A coin is dropped from a window. Find the coin's
(a) average speed during the first 4 sec of fall and
(b) instantaneous speed at t=4 .

Question 6

Determine
if the limit exists.

A) -1
B)
C)
D) 1
E) The limit does not exist.

Question 7

What is the instantaneous rate of change of at x=1 ?

A) 2
B) 0
C) 4
D) 1

Question 8

For the function
(a) Find the slope of the function at x=1 using the definition of the derivative.
(b) What is the equation of the tangent line to the curve f(x) at x=1 ?

Question 9

The following table lists the population statistics for a certain city.

What is the average rate of change in the population with respect to time between 1975 and 2005 ?

Question 10

For the function f(x)=4+3 x-2 x² :
(a) Use the definition of the derivative at a point to find the slope of the tangent line of the function at x=1 .
(b) What is the equation of the tangent line in part a?

Question 11

Write the integral that would be used to find the shaded area on the graph shown below.

Question 12

What is the average rate of change of over the interval [1.9,2.1] ?

Question 13

Use the graph below to find the limits or explain why the limits do not exist.
(a)
(b)
(c)

Question 14

Compute the integral by computing a geometric area. (Hint: Graph the function first.)

Question 15

Find the derivative of f(x)=7 x+10 .

Question 16

Draw the graph of over the interval [0,4] .

On the graph, show and shade the rectangles that would be used to approximate the area under the curve by the right rectangle approximation method using 4 subintervals.

Question 17

Draw the graph of f(x)=-2(x-1)(x-4) over the interval [1,4] . On the graph, show and shade the rectangles that would be used to approximate the area under the curve f(x) over [1,4] by the right rectangle approximation method using 6 subintervals. Compute the estimation.

Question 18

At what points c in the domain of exist?

Question 19

Find the average rate of change of f(x)=3 \cos x over the interval

A) -3
B)
C)
D)

Question 20

Partition the interval [1,5] into eight equal subintervals. List the eight subintervals.

Question 21

Question 22

Explain how to find the area under the graph of from x=0 to x=4 by computing the geometric area.

Question 23

Find the derivative of f(x)=-4 x-8 .

Question 24

Use a calculator to find the RRAM area approximation for the area under the graph from x=0 to x=2 with 20 approximating rectangles.

Question 25

Determine
if the limit exists.

A) -1
B)
C)
D) 1
E) The limit does not exist.

Question 26

For the function
(a) Find the slope of the function at x=2 using the definition of the derivative.
(b) What is the equation of the tangent line to the curve f(x) at x=2 ?

Question 27

Find the instantaneous rate of change of at x=4 .

Question 28

Explain how to find the area under the graph of from x=1 to x=5 by computing the geometric area.

Question 29

Use a calculator to find the LRAM area approximation for the area under the graph
from x=0 to x=2 with 20 approximating rectangles.

Question 30

(a) Explain why direct substitution cannot be used to find the limit.
(b) Find the limit algebraically, if it exists.

Question 31

Determine the following limits, if they exist.
(a)
(b)
(c)
(d)
(e)
(f)

Question 32

Write the integral that would be used to find the shaded area on the graph shown below.

Question 33

At what points c in the domain of f(x) does exist?

Question 34

Find the average rate of change of f(x)=5 \tan x over the interval

A)
B)
C) -10
D) 0

Question 35

Use the graph below to find the limits or explain why the limits do not exist.
(a)
(b)
(c)

Question 36

A baseball is dropped from a window. Find the baseball's (a) average speed during the first 2 sec of fall and (b) instantaneous speed at t=2 .

Question 37

The following table lists the population statistics for a certain city.

What is the average rate of change in the population with respect to time between 1975 and 2005 ?