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

2.3 Di�erentiation Formulas (Homework)

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2021/9/18 下午10:05 2.3 Differentiation Formulas – Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021 | WebAssign

https://www.webassign.net/web/Student/Assignment-Responses/last?dep=27128647 2/20

Differentiate the function.

Differentiate the function.

Differentiate the function.

g(t) = 7t + 9t2

g ′(t) =

g(x) = x − 2x + 11 9 4

2

g ′(x) =

V(t) = t + t−3 ⁄ 5 5

V ′(t) =

1. [–/1 Points] SCALC9 2.3.002.DETAILS

2. [–/1 Points] SCALC9 2.3.004.MI.DETAILS

3. [–/1 Points] SCALC9 2.3.008.DETAILS

2021/9/18 下午10:05 2.3 Differentiation Formulas – Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021 | WebAssign

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Differentiate the function.

Differentiate the function.

Differentiate the function after first rewriting the function in a different form. (Do not use the product or quotient rules.)

h(w) = w − 6 6

h ′(w) =

S(R) = 9 R2

S'(R) =

F(t) = (5t − 4)2

F ′(t) =

4. [–/1 Points] SCALC9 2.3.012.DETAILS

5. [–/1 Points] SCALC9 2.3.014.DETAILS

6. [–/1 Points] SCALC9 2.3.016.DETAILS

2021/9/18 下午10:05 2.3 Differentiation Formulas – Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021 | WebAssign

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Differentiate the function after first rewriting the function in a different form. (Do not use the product or quotient rules.)

Differentiate the function after first rewriting the function in a different form. (Do not use the product or quotient rules.)

=

y = 7 + 7x

2x

x

2

y ′ =

G(t) = + 6t t

3

G ′(t)

7. [–/1 Points] SCALC9 2.3.018.DETAILS

8. [–/1 Points] SCALC9 2.3.020.DETAILS

2021/9/18 下午10:05 2.3 Differentiation Formulas – Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021 | WebAssign

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Differentiate the function after first rewriting the function in a different form. (Do not use the product or quotient rules.)

Consider the following.

Find

Find

F(z) = A + Bz + Cz

z

2

2

F ′(z) =

y = + t

x8 x t

. dy dx

=

dy dx

. dy dt

=

dy dt

10. [–/2 Points] SCALC9 2.3.026.DETAILS

2021/9/18 下午10:05 2.3 Differentiation Formulas – Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021 | WebAssign

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Find the derivative of the function

in two ways: by using the Quotient Rule and by simplifying first. (Check that your answers are equivalent. Which method do you prefer?)

Quotient Rule =

simplify first =

Use the quotient rule to find the derivative of the function.

F(x) = x − 7x +

x

4 3 x

2

F ′(x)

F ′(x)

G(u) = 5u − 4u u + 1

4

G ′(u) =

11. [–/2 Points] SCALC9 2.3.028.DETAILS

12. [–/1 Points] SCALC9 2.3.036.DETAILS

2021/9/18 下午10:05 2.3 Differentiation Formulas – Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021 | WebAssign

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Differentiate.

Differentiate.

F(x) = 1

3x − 8x + 93 2

F ′(x) =

y = + 7

x

x

y′ =

13. [–/1 Points] SCALC9 2.3.038.DETAILS

14. [–/1 Points] SCALC9 2.3.040.DETAILS

2021/9/18 下午10:05 2.3 Differentiation Formulas – Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021 | WebAssign

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Differentiate.

Differentiate.

Differentiate.

h(w) = (w + 6w )(w − w )5 4 −4 −6

h ′(w) =

F(t) = At

Bt + Ct

2

6 9

F ′(t) =

f(x) = ax + b cx + d

f ‘(x) =

15. [–/1 Points] SCALC9 2.3.046.DETAILS

16. [–/1 Points] SCALC9 2.3.050.DETAILS

2021/9/18 下午10:05 2.3 Differentiation Formulas – 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.

Consider the following.

Find an equation of the tangent line.

y =

Find an equation of the normal line.

y =

y = 3x − x + 6, (2, 26)3 2

y =

y = x , (1, 1)5 ⁄ 4

18. [–/1 Points] SCALC9 2.3.060.DETAILS

19. [–/2 Points] SCALC9 2.3.062.DETAILS

2021/9/18 下午10:05 2.3 Differentiation Formulas – Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021 | WebAssign

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Find equations of the tangent line and normal line to the curve at the specified point.

tangent y =

normal y =

y = , 1, 0.2 x + 4

x

20. [–/2 Points] SCALC9 2.3.064.DETAILS

2021/9/18 下午10:05 2.3 Differentiation Formulas – Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021 | WebAssign

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A graphing calculator is recommended.

The graph of the function is called a serpentine.

(a) Find an equation of the tangent line to this curve at the point

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

f(x) = x

1 + x2

(3, 0.30).

y =

21. [–/2 Points] SCALC9 2.3.066.DETAILS

2021/9/18 下午10:05 2.3 Differentiation Formulas – Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021 | WebAssign

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Find the first and second derivative of the function.

=

=

G(r) = + r r 7

G'(r)

G”(r)

22. [–/2 Points] SCALC9 2.3.068.DETAILS

2021/9/18 下午10:05 2.3 Differentiation Formulas – Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021 | WebAssign

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The equation of motion of a particle is where s is in meters and t is in seconds. (Assume t ≥ 0.)

(a) Find the velocity, v(t), and acceleration, a(t), as functions of t.

v(t) =

a(t) =

(b) Find the acceleration (in m/s ) after 1.3 s.

(c) Graph the position, velocity, and acceleration functions on the same screen.

s = t − 4t + t − t,4 3 2

2

a(1.3) = m/s2

23. [–/4 Points] SCALC9 2.3.074.DETAILS

2021/9/18 下午10:05 2.3 Differentiation Formulas – Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021 | WebAssign

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2021/9/18 下午10:05 2.3 Differentiation Formulas – Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021 | WebAssign

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2021/9/18 下午10:05 2.3 Differentiation Formulas – Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021 | WebAssign

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The number of tree species S in a given area A in a forest reserve has been modeled by the power function

where A is measured in square meters. Find (Round your answer to five decimal places.)

The rate of change of the number of tree species with respect to area is 0.35599 species/m .

The rate of change of the number of tree species with respect to area is −44.39254 species/m .

The rate of change of the number of tree species with respect to area is −0.35599 species/m .

The rate of change of the number of tree species with respect to area is 44.39254 species/m .

The rate of change of the number of tree species with respect to area is 44.03655 species/m .

Suppose that and Find

(a)

(b)

(c)

(d)

S(A) = 0.882A0.842

S'(105).

S'(105) = species/m2

2

2

2

2

2

f(4) = 4, g(4) = 5, f ′(4) = −3, g ′(4) = 2. h ′(4).

h(x) = 2f(x) + 3g(x)

h ′(4) =

h(x) = f(x)g(x)

h ′(4) =

h(x) = f(x) g(x)

h ′(4) =

h(x) = g(x)

f(x) + g(x)

h ′(4) =

24. [–/2 Points] SCALC9 2.3.076.DETAILS

25. [–/4 Points] SCALC9 2.3.080.DETAILS

2021/9/18 下午10:05 2.3 Differentiation Formulas – Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021 | WebAssign

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If find

Let and where F and G are the functions whose graphs are shown.

(a) Find

(b) Find

h(2) = 3 and h'(2) = −7,

. d dx

h(x) x

x = 2

P(x) = F(x)G(x) Q(x) = , F(x) G(x)

P'(2).

Q'(7).

26. [–/1 Points] SCALC9 2.3.082.DETAILS

2021/9/18 下午10:05 2.3 Differentiation Formulas – Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021 | WebAssign

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If f is a differentiable function, find an expression for the derivative of each of the following functions.

(a)

(b)

(c)

(d)

y = x f(x)6

y’ = x f ‘(x) + 6x f(x)6 5

y’ = x f ‘(x) − 5x f(x)5 6

y’ = x f(x) − 6x f ‘(x)6 5

y’ = x f(x) + 6x f ‘(x)6 5

y’ = x f ‘(x) + 5x f(x)5 6

y = f(x)

x5

y’ = xf ‘(x) + 4f(x)

x5

y’ = xf(x) + 5f ‘(x)

x5

y’ = xf ‘(x) − 5f(x)

x6

y’ = xf ‘(x) − 4f(x)

x6

y’ = xf(x) − 5f ‘(x)

x4

y = x f(x)

4

y’ = f ‘(x)(3x ) − x f ‘(x)

[f(x)]

3 4

2

y’ = f ‘(x)(4x ) − x f(x)

[f(x)]

3 4

2

y’ = f ‘(x)(3x ) + x f(x)

[f(x)]

4 3

2

y’ = f(x)(3x ) + x f ‘(x)

[f(x)]

4 4

2

y’ = f(x)(4x ) − x f ‘(x)

[f(x)]

3 4

2

y = 7 + xf(x)

x

y’ = xf(x) + 7x f(x) − 2

2x

2

3/2

y’ = xf ‘(x) − 2x f ‘(x) + 7

2x

2

3/2

28. [–/4 Points] SCALC9 2.3.086.DETAILS

2021/9/18 下午10:05 2.3 Differentiation Formulas – Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021 | WebAssign

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For what values of x does the graph of f(x) have a horizontal tangent? (Enter your answers as a comma-separated list.)

Find equations of the tangent lines to the curve

that are parallel to the line (Enter your answers as a comma-separated list of equations.)

y’ = xf ‘(x) + 7x f(x) − 2

2x

2

3/2

y’ = xf(x) + 2x f ‘(x) − 7

2x

2

3/2

y’ = xf(x) − 7x f ‘(x) + 7

2x

2

3/2

f(x) = x + 3x + x + 73 2

x =

y = x − 1 x + 1

x − 2y = 5.

29. [–/1 Points] SCALC9 2.3.088.MI.DETAILS

30. [–/1 Points] SCALC9 2.3.092.DETAILS

2021/9/18 下午10:05 2.3 Differentiation Formulas – Fal21 MATH 005A #71943 SINGLE VARIABLE CALCULUS I, Fall 2021 | WebAssign

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Find equations of both lines through the point (2, −3) that are tangent to the parabola

smaller slope y =

larger slope y =

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