Mathematics assignment

Kyle Taitt CSU Webwork

MATH 160 WeBWorK assignment M160-801-FA-2.1 due 09/23/2016 at 11:59pm MDT

1. (1 point) The experimental data in the table below define y as a function of x.

x 0 1 2 3 4 5 y 2.8 1.7 1.4 1.8 2.6 3.6

A. Let P be the point (2, 1.4). Find the slopes of the secant lines PQ when Q is the point of the graph with x coordinate x1.

x1 0 1 3 4 5 slope

B. Draw the graph of the function for yourself and estimate the slope of the tangent line at P.

2. (1 point) An automobile starts from rest and travels down a straight section of road. The distance s (in feet) of the car from the starting position after t seconds is given by s(t) = 4t3.

(a) Find the instantaneous velocity (in feet per second) at t = 3 seconds.

Instantaneous velocity at t = 3 is ft/s. (b) Find the instantaneous velocity in (feet per second) at

t = 8 seconds. Instantaneous velocity at t = 8 is ft/s.

3. (1 point) Suppose that an object moves along an s-axis so that its location is given by s(t) = t2 +8t at time t. (Here s is in meters and t is in seconds.)

(a) Find the average velocity of the object in meters per sec- ond over the time interval t = 3 to t = 9 seconds.

Average velocity = m/s. (b) Find the instantaneous velocity of the object in meters per

second at t = 5 seconds. Instantaneous velocity = m/s.

4. (1 point) The point P(4,4) lies on the curve y = p

x+ 2. Let Q be the point (x,

p x+2).

a.) Find the slope of the secant line PQ for the following values of x. (Answers here should be correct to at least 6 places after the decimal point.)

If x = 4.1, the slope of PQ is: If x = 4.01, the slope of PQ is: If x = 3.9, the slope of PQ is: If x = 3.99, the slope of PQ is:

b.) Based on the above results, estimate the slope of the tangent line to the curve at P(4,4).

Answer:

5. (1 point) If a ball is thrown straight up into the air with an initial velocity of 100 ft/s, it height in feet after t second is given by y = 100t �16t2. Find the average velocity for the time period begining when t = 2 and lasting (i) 0.1 seconds

(ii) 0.01 seconds

(iii) 0.001 seconds

Finally based on the above results, guess what the instanta- neous velocity of the ball is when t = 2.

6. (1 point) Use the definition of the derivative at a point to find the slope of the tangent line to y =

p 2×2 �2 at the point

x = 4. m = What is the equation of this tangent line, written in point-slope form? y = + (x� )

7. (1 point) This problem finds the derivative of a function at a point using the formal definition. The process is broken down into the following steps: Let f (x) be the function 8×2 �2x+10. Then the quotient f (7+h)� f (7)

h

can be simplified to ah+b for: a = and b = Use a limit as h ! to calculate f 0(7) =

8. (1 point) This problem finds the derivative of a function at a point using the formal definition. The process is broken down into the following steps: Let f (x) be the function 1

x+6 . Then the quotient f (7+h)� f (7)

h

can be simplified to �1

ah+b for: a = and b = Use a limit as h ! to calculate f 0(7) = .

1

9. (1 point) Let f (t) be the function 4t � 10 t

.

Use the formal definition of the derivative lim h!0

f (t +h)� f (t) h

to find d f

dt

.

Enter the simplified difference quotient f (t +h)� f (t)

h

(BE- FORE computing the limit.)

Enter your answer for d f

dt

(AFTER computing the limit.)

10. (1 point) Let f (q) be the function 7p

q+10 .

Use the formal definition of the derivative lim h!0

f (q+h)� f (q) h

to find d f

dq

.

Enter the simplified difference quotient f (q+h)� f (q)

h

(BE- FORE computing the limit.)

(Note: simplify to the point where you can evaluate the limit algebraically, i.e. until you are no longer dividing by 0.)

Enter your answer for d f

dq

(AFTER computing the limit.)

11. (1 point) Let f (s) be the function 11s2 +4s+4. Follow the steps below to use the formal definition of the deriv-

ative lim h!0

f (s+h)� f (s) h

to find d f

ds

.

Substitute and evaluate: f (s+h) = Simplify f (s+h)� f (s) =

Enter the simplified difference quotient f (s+h)� f (s)

h

(BE- FORE computing the limit.)

(Note: simplify to the point where you could evaluate the limit algebraically, i.e. until you are no longer di- viding by 0.)

Enter your answer for d f

ds

(AFTER computing the limit.)

12. (1 point) Let f (t) be the function 10�t t

. Follow the steps below to use the formal definition of the deriv-

ative lim h!0

f (t +h)� f (t) h

to find d f

dt

.

Substitute and evaluate: f (t +h) =

Simplify f (t +h)� f (t) =

Enter the simplified difference quotient f (t +h)� f (t)

h

(BE- FORE computing the limit.)

(Note: simplify to the point where you could evaluate the limit algebraically, i.e. until you are no longer di- viding by 0.)

Enter your answer for d f

dt

(AFTER computing the limit.)

13. (1 point) Let f (t) be the function p

9t. Follow the steps below to use the formal definition of the deriv-

ative lim h!0

f (t +h)� f (t) h

to find d f

dt

.

Substitute and evaluate: f (t +h) = Simplify f (t +h)� f (t) =

Enter the simplified difference quotient f (t +h)� f (t)

h

(BE- FORE computing the limit.)

(Note: simplify to the point where you could evaluate the limit algebraically, i.e. until you are no longer di- viding by 0.)

Enter your answer for d f

dt

(AFTER computing the limit.)

14. (1 point)

Which graph from A-D below has as its derivative the graph above? [?/A/B/C/D]

2

A B

C D

(Click on a graph to enlarge it.)

15. (1 point)

Which graph from A-D below has as its derivative the graph above? [?/A/B/C/D]

A B

C D

(Click on a graph to enlarge it.)

16. (1 point)

Let

f (x) =

( x

2 +1 if x < 1 x+1 if x � 1

Calculate the left-hand derivative: lim

h!0� f (1+h)� f (1)

h

=

Calculate the right-hand derivative: 3

lim h!0+

f (1+h)� f (1) h

=

Is f (x) differentiable at x = 1? (Yes or No)

17. (1 point) Let f (x) = x3 �13x. Calculate the difference quotient f (2+h)� f (2)

h

for h = .1 h = .01 h =�.01 h =�.1 If someone now told you that the derivative (slope of the tangent line to the graph) of f (x) at x = 2 was an integer, what would you expect it to be?

18. (1 point) Let f (x) = 2

x�8 . Then according to the defini- tion of derivative

f

0(x) = lim t!x

(Your answer above and the next few answers below will in- volve the variables t and x.)

NOTE: In this question, webwork will only grade an answer once you have entered BOTH the numerator and the denomina- tor. Until then, it will mark answers as wrong. The expression inside the limit simplifies to a simple fraction with:

Numerator =

Denominator =

We can cancel the factor appearing in the de- nominator against a similar factor appearing in the numerator leaving a simpler fraction with:

Numerator =

Denominator =

Taking the limit of this fractional expression gives us f

0(x) =

19. (1 point) The graph shown is the graph of the SLOPE of the tangent line of the original function. (This slope is also called the derivative of f.) For each interval, enter all letters whose corresponding state- ments are true for that interval.

1. The interval from a to b 2. The interval from b to c 3. The interval from c to d 4. The interval from d to e 5. The interval from e to f

A. The slope of the original function is positive on this in- terval

B. The slope of the original function is negative on this interval.

C. The slope of the original function is increasing on this interval.

D. The slope of the original function is decreasing on this interval.

E. The original function is increasing on this interval. F. The original function is decreasing on this interval. G. The shape of the original function is concave up on this

interval. H. The shape of the original function is concave down on

this interval.

20. (1 point) Cups are filled with coffee. Match the cup shapes with the graphs showing the height of the fluid level as a function of time.

4

? 1.

? 2.

? 3.

? 4.

A B

C D

(Click on a graph to enlarge it)

5

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