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2021/9/18 下午2:53 2.1 Derivatives and Rates of Change - Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021 | WebAssign

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Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021

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2021/9/18 下午2:53 2.1 Derivatives and Rates of Change - Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021 | WebAssign

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2021/9/18 下午2:53 2.1 Derivatives and Rates of Change - Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021 | WebAssign

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(a) Find the slope of the tangent line to the parabola at the point by using the following parameters.

(i) The tangent line to the curve at the point is the line through P with slope

provided that this limit exists.

m =

(ii) The expression of slope is

m =

(b) Find an equation of the tangent line in part (a).

y =

(c) Graph the parabola and the tangent line. As a check on your work, zoom in toward the point until the parabola and the tangent line are indistinguishable.

y = x + 6x2 (−2, −8)

y = f(x) P(a, f(a))

m = lim x→a

f(x) − f(a) x − a

m = .lim h→0

f(a + h) − f(a) h

(−2, −8)

ℹ

ℹ

1. [–/4 Points] SCALC9 2.1.003.DETAILS

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2021/9/18 下午2:53 2.1 Derivatives and Rates of Change - Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021 | WebAssign

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Find an equation of the tangent line to the curve at the given point.

y =

ℹ

ℹ

y = , (−1, 6)3 − 33x

2. [–/1 Points] SCALC9 2.1.008.DETAILS

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2021/9/18 下午2:53 2.1 Derivatives and Rates of Change - Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021 | WebAssign

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If a rock is thrown upward on the planet Mars with a velocity of its height above the ground (in meters) after t seconds is given by

(a) Find the velocity (in m/s) of the rock after 2 seconds.

m/s

(b) Find the velocity (in m/s) of the rock when

m/s

(c) When (in seconds) will the rock hit the surface? (Round your answer to one decimal place.)

t = s

(d) With what velocity (in m/s) will the rock hit the surface?

m/s

13 m/s, H = 13t − 1.86t .2

t = a.

3. [–/4 Points] SCALC9 2.1.012.MI.DETAILS

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2021/9/18 下午2:53 2.1 Derivatives and Rates of Change - Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021 | WebAssign

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(a) Find the slope, m, of the tangent to the curve at the point where

(b) Find equations of the tangent lines at the following points.

(25, 10)

y =

(4, 4)

y =

(c) Graph the curve and both tangents on a common screen.

y = 2 x x = a.

m =

ℹ

ℹ

4. [–/4 Points] SCALC9 2.1.010.DETAILS

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2021/9/18 下午2:53 2.1 Derivatives and Rates of Change - Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021 | WebAssign

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ℹ

ℹ

2021/9/18 下午2:53 2.1 Derivatives and Rates of Change - Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021 | WebAssign

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The following graph shows the position functions of two runners, A and B, who run a 100-meter race and finish in a tie.

(a) Describe and compare how the runners run the race.

Runner A starts the race at a faster velocity than runner B but finishes the race at a slower velocity. Runner B runs the entire race at the same velocity.

Runner A and runner B both start and end the race a the same velocities.

Runner A runs the entire race at the same velocity. Runner B starts the race at a faster velocity than runner A but finishes the race at a slower velocity.

Runner A runs the entire race at the same velocity. Runner B starts the race at a slower velocity than runner A but finishes the race at a faster velocity.

Runner A starts the race at a slower velocity than runner B but finishes the race at a faster velocity. Runner B runs the entire race at the same velocity.

(b) How long after the race began (in seconds) is the distance between the runners the greatest?

s

(c) How long after the race began (in seconds) do the runners have the same velocity?

s

ℹ

5. [–/3 Points] SCALC9 2.1.016.DETAILS MY NOTES ASK YOUR TEACHER

2021/9/18 下午2:53 2.1 Derivatives and Rates of Change - Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021 | WebAssign

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The displacement (in feet) of a particle moving in a straight line is given by where t is measured in

seconds.

(a) Find the average velocity (in ft/s) over each time interval.

(i) [4, 8]

ft/s

(ii) [6, 8]

ft/s

(iii) [8, 10]

ft/s

(iv) [8, 12]

ft/s

(b) Find the instantaneous velocity (in ft/s) when

ft/s

(c) Draw the graph of s as a function of t and draw the secant lines whose slopes are the average velocities in part (a). Then select the graph of s with the tangent line whose slope is the instantaneous velocity in part (b).

s = t − 5t + 17, 1 2

2

t = 8.

6. [–/6 Points] SCALC9 2.1.014.DETAILS

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2021/9/18 下午2:53 2.1 Derivatives and Rates of Change - Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021 | WebAssign

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ℹ

ℹ

2021/9/18 下午2:53 2.1 Derivatives and Rates of Change - Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021 | WebAssign

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ℹ

ℹ

2021/9/18 下午2:53 2.1 Derivatives and Rates of Change - Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021 | WebAssign

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The graph of a function f is shown.

(a) Find the average rate of change of f on the interval

(b) Identify an interval on which the average rate of change of f is 0.

[10, 50]

[0, 80]

[10, 40]

[0, 60]

[20, 40]

(c) Compute the following.

What does this value represent geometrically?

the slope of the line segment from (0, f(0)) to (40, f(40))

the slope of the tangent line at (40, f(40))

the slope of the tangent line at (0, f(0))

the slope of the tangent line at (20, f(20))

(d) Estimate the value of

(e) Is

Yes

No

(f) Is

ℹ

[50, 60].

f(40) − f(0) 40 − 0

f' (50).

f' (10) > f' (30)?

f' (60) > ? f(80) − f(40)

80 − 40

7. [–/8 Points] SCALC9 2.1.018.DETAILS

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2021/9/18 下午2:53 2.1 Derivatives and Rates of Change - Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021 | WebAssign

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Yes

No

Explain.

The slope of the tangent line at x = 60, f ′(60), is greater than the slope of the line passing through (40, f(40)) and (80, f(80)).

The slope of the tangent line at x = 60, f ′(60), is less than the slope of the line passing through (40, f(40)) and (80, f(80)).

The slope of the tangent line at x = 80, f ′(80), is greater than the slope of the line passing through (40, f(40)) and (60, f(60)).

The slope of the tangent line at x = 40, f ′(40), is less than the slope of the line passing through (60, f(60)) and (80, f(80)).

The slope of the tangent line at x = 80, f ′(80), is less than the slope of the line passing through (40, f(40)) and (60, f(60)).

Use this definition to find f ′(a) at the given number a.

Find f ′(a).

f(x) = 7x , a = −14

f(t) = t − 8t3

8. [–/1 Points] SCALC9 2.1.020.DETAILS

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9. [–/1 Points] SCALC9 2.1.024.DETAILS

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2021/9/18 下午2:53 2.1 Derivatives and Rates of Change - Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021 | WebAssign

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Find an equation of the tangent line to the graph of (Enter your answer as an equation in terms of y and x.)

y = g(x) at x = 3 if g(3) = −6 and g'(3) = 2.

10. [–/1 Points] SCALC9 2.1.028.DETAILS

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2021/9/18 下午2:53 2.1 Derivatives and Rates of Change - Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021 | WebAssign

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A graphing calculator is recommended.

(a) If find and use it to find equations of the tangent lines to the curve at the points and

=

passing through (3, 18) =

passing through (4, 16) =

(b) Illustrate part (a) by graphing the curve and the tangent lines on the same screen.

G(x) = 5x − x ,2 3 G′(a) y = 5x − x2 3 (3, 18) (4, 16).

G'(a)

y (x)1

y (x)2

11. [–/4 Points] SCALC9 2.1.032.DETAILS

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2021/9/18 下午2:53 2.1 Derivatives and Rates of Change - Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021 | WebAssign

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ℹ

ℹ

2021/9/18 下午2:53 2.1 Derivatives and Rates of Change - Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021 | WebAssign

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ℹ

ℹ

2021/9/18 下午2:53 2.1 Derivatives and Rates of Change - Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021 | WebAssign

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If the tangent line to at (4, 3) passes through the point (0, 2), find and

=

=

A particle moves along a straight line with equation of motion s = f(t), where s is measured in meters and t in seconds. Find the velocity and speed (in m/s) when t = 4.

velocity m/s

speed m/s

y = f(x) f(4) f ′(4).

f(4)

f '(4)

f(t) = 16 + 30

t + 1

12. [–/2 Points] SCALC9 2.1.034.DETAILS

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13. [–/2 Points] SCALC9 2.1.036.DETAILS

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2021/9/18 下午2:53 2.1 Derivatives and Rates of Change - Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021 | WebAssign

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A roast turkey is taken from an oven when its temperature has reached 185°F and is placed on a table in a room where the temperature is 75°F. The graph shows how the temperature of the turkey decreases and gradually approaches room temperature.

By measuring the slope of the tangent, estimate the rate of change of the temperature after an hour (in °F/min). (Round your answer to one decimal place.)

°F/min

ℹ

14. [–/1 Points] SCALC9 2.1.038.DETAILS MY NOTES ASK YOUR TEACHER

2021/9/18 下午2:53 2.1 Derivatives and Rates of Change - Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021 | WebAssign

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The limit represents the derivative of some function f at some number a. State such an f and a.

lim

→ 4

cos( ) −

−

2

2

4

f(x) = sin(x), a = 3

f(x) = sin(x), a = 6

f(x) = cos(x), a = 6

f(x) = sin(x), a = 4

f(x) = cos(x), a = 3

f(x) = cos(x), a = 4

15. [–/1 Points] SCALC9 2.1.048.DETAILS

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The quantity (in pounds) of a gourmet ground coffee that is sold by a coffee company at a price of p dollars per pound is

(a) What is the meaning of the derivative What are its units?

The rate of change of the price per pound with respect to the quantity of coffee sold when the price is $7 per pound. The units are pounds/(dollars/pound).

The rate of change of the quantity of coffee sold with respect to the price per pound when the price is $7 per pound. The units are pounds/(dollars/pound).

The rate of change of the price per pound with respect to the quantity of coffee sold when the price is $7 per pound. The units are dollars/pound.

The price of the coffee as a function of the supply. The units are pounds.

The rate of change of the quantity of coffee sold with respect to the price per pound when the price is $7 per pound. The units are dollars/pound.

(b) Is positive or negative? Explain.

positive

negative

Q = f(p).

f ′(7)?

f ′(7)

16. [–/2 Points] SCALC9 2.1.052.DETAILS

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The graph shows the influence of the temperature T on the maximum sustainable swimming speed S of Coho salmon.

(a) What is the meaning of the derivative What are its units?

S′(T) is the rate of change of the maximum sustainable speed of Coho salmon with respect to time. The units are cm/s .

S′(T) is the rate of change of the temperature with respect to the maximum sustainable speed of Coho salmon. The units are °C/(cm/s).

S′(T) is the maximum sustainable speed of Coho salmon for a given temperature. The units are (cm/s).

S′(T) is the rate of change of the temperature with respect to the time. The units are °C/s.

S′(T) is the rate of change of the maximum sustainable speed of Coho salmon with respect to the temperature. The units are (cm/s)/°C.

(b) Estimate the values of and and interpret them.

≈

≈

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ℹ

S′(T)?

2

S′(15) S′(25)

S'(15)

S'(25)

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17. [–/3 Points] SCALC9 2.1.054.DETAILS

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