시험 결과 - Stewart 9th Section 2.3: Differentiation Formulas

1 Differentiation - Basic 오답률 100%

오답

$ g(x) = 4x + 7 $

(미작성)

정답

$g'(x) = 4$

위 정답과 비교하여 채점하세요:

해설 없음

2 Differentiation - Basic 오답률 100%

오답

$ g(t) = 5t + 4t^2 $

(미작성)

정답

$g'(t) = 5 + 8t$

위 정답과 비교하여 채점하세요:

해설 없음

3 Differentiation - Basic 오답률 100%

오답

$ f(x) = x^{75} - x + 3 $

(미작성)

정답

$f'(x) = 75x^{74} - 1$

위 정답과 비교하여 채점하세요:

해설 없음

4 Differentiation - Basic 오답률 100%

오답

$ g(x) = \dfrac{7}{4} x^2 - 3x + 12 $

(미작성)

정답

$g'(x) = \dfrac{7}{2} x - 3$

위 정답과 비교하여 채점하세요:

해설 없음

5 Differentiation - Basic 오답률 100%

오답

$ W(v) = 1.8 v^{-3} $

(미작성)

정답

$W'(v) = -5.4 v^{-4}$

위 정답과 비교하여 채점하세요:

해설 없음

6 Differentiation - Basic 오답률 100%

오답

$ r(z) = z^{-5} - z^{\dfrac{1}{2}} $

(미작성)

정답

$r'(z) = -5z^{-6} - \dfrac{1}{2} z^{\dfrac{-1}{2}}$

위 정답과 비교하여 채점하세요:

해설 없음

7 Differentiation - Basic 오답률 100%

오답

$ f(x) = x^{\dfrac{3}{2}} + x^{-3} $

(미작성)

정답

$f'(x) = \dfrac{3}{2} x^{\dfrac{1}{2}} - 3x^{-4}$

위 정답과 비교하여 채점하세요:

해설 없음

8 Differentiation - Basic 오답률 100%

오답

$ V(t) = t^{\dfrac{-3}{5}} + t^4 $

(미작성)

정답

$V'(t) = \dfrac{-3}{5} t^{\dfrac{-8}{5}} + 4t^3$

위 정답과 비교하여 채점하세요:

해설 없음

9 Differentiation - Basic 오답률 100%

오답

$ s(t) = \dfrac{1}{t} + \dfrac{1}{t^2} $

(미작성)

정답

$s'(t) = -\dfrac{1}{t^2} - \dfrac{2}{t^3}$

위 정답과 비교하여 채점하세요:

해설 없음

10 Differentiation - Basic 오답률 100%

오답

$ r(t) = \dfrac{a}{t^2} + \dfrac{b}{t^4} $

(미작성)

정답

$r'(t) = \dfrac{-2a}{t^3} + \dfrac{-4b}{t^5}$

위 정답과 비교하여 채점하세요:

해설 없음

11 Differentiation - Basic 오답률 100%

오답

$ y = 2x + \sqrt{x} $

(미작성)

정답

$y' = 2 + \dfrac{1}{2 \sqrt{x}}$

위 정답과 비교하여 채점하세요:

해설 없음

12 Differentiation - Basic 오답률 100%

오답

$ h(w) = \sqrt{2} w - \sqrt{2} $

(미작성)

정답

$h'(w) = \sqrt{2}$

위 정답과 비교하여 채점하세요:

해설 없음

13 Differentiation - Basic 오답률 100%

오답

$ g(x) = \dfrac{1}{\sqrt{x}} + \sqrt[5]{x} $

(미작성)

정답

$g'(x) = -\dfrac{1}{2} x^{\dfrac{-3}{2}} + \dfrac{1}{5} x^{\dfrac{-4}{5}}$

위 정답과 비교하여 채점하세요:

해설 없음

14 Differentiation - Basic 오답률 100%

오답

$ S(R) = 4 \pi R^2 $

(미작성)

정답

$S'(R) = 8 \pi R$

위 정답과 비교하여 채점하세요:

해설 없음

15 Differentiation - Simplify First 오답률 100%

오답

$ f(x) = x^3 (x + 3) $

(미작성)

정답

$f'(x) = 4x^3 + 9x^2$

위 정답과 비교하여 채점하세요:

해설 없음

16 Differentiation - Simplify First 오답률 100%

오답

$ F(t) = (2t - 3)^2 $

(미작성)

정답

$F'(t) = 8t - 12$

위 정답과 비교하여 채점하세요:

해설 없음

17 Differentiation - Simplify First 오답률 100%

오답

$ f(x) = \dfrac{3x^2 + x^3}{x} $

(미작성)

정답

$f'(x) = 3 + 2x$

위 정답과 비교하여 채점하세요:

해설 없음

18 Differentiation - Simplify First 오답률 100%

오답

$ y = \dfrac{\sqrt{x} + x}{x^2} $

(미작성)

정답

$y' = -\dfrac{3}{2} x^{\dfrac{-5}{2}} - x^{-2}$

위 정답과 비교하여 채점하세요:

해설 없음

19 Differentiation - Simplify First 오답률 100%

오답

$ G(q) = (1 + q^{-1})^2 $

(미작성)

정답

$G'(q) = -2q^{-2} - 2q^{-3}$

위 정답과 비교하여 채점하세요:

해설 없음

20 Differentiation - Simplify First 오답률 100%

오답

$ G(t) = \sqrt{5t} + \dfrac{\sqrt{7}}{t} $

(미작성)

정답

$G'(t) = \dfrac{\sqrt{5}}{2 \sqrt{t}} - \dfrac{\sqrt{7}}{t^2}$

위 정답과 비교하여 채점하세요:

해설 없음

21 Differentiation - Simplify First 오답률 100%

오답

$ G(r) = \dfrac{3r^{\dfrac{3}{2}} + r^{\dfrac{5}{2}}}{r} $

(미작성)

정답

$G'(r) = \dfrac{3}{2} r^{\dfrac{-1}{2}} + \dfrac{3}{2} r^{\dfrac{1}{2}}$

위 정답과 비교하여 채점하세요:

해설 없음

22 Differentiation - Simplify First 오답률 100%

오답

$ F(z) = \dfrac{A + B z + C z^2}{z^2} $

(미작성)

정답

$F'(z) = -2A z^{-3} - B z^{-2}$

위 정답과 비교하여 채점하세요:

해설 없음

23 Differentiation - Simplify First 오답률 100%

오답

$ P(w) = \dfrac{2w^2 - w + 4}{\sqrt{w}} $

(미작성)

정답

$P'(w) = 3w^{\dfrac{1}{2}} - \dfrac{1}{2} w^{\dfrac{-1}{2}} - 2w^{\dfrac{-3}{2}}$

위 정답과 비교하여 채점하세요:

해설 없음

24 Differentiation - Simplify First 오답률 100%

오답

$ D(t) = \dfrac{1 + 16t^2}{(4t)^3} $

(미작성)

정답

$D'(t) = -\dfrac{3}{64} t^{-4} + \dfrac{1}{4} t^{-2}$

위 정답과 비교하여 채점하세요:

해설 없음

25 Differentiation - Simplify First 오답률 100%

오답

Find $\dfrac{d y}{d x}$ and $\dfrac{d y}{d t}$. $y = t x^2 + t^3 x$

(미작성)

정답

$\dfrac{d y}{d x} = 2t x + t^3$, $\dfrac{d y}{d t} = x^2 + 3t^2 x$

위 정답과 비교하여 채점하세요:

해설 없음

26 Differentiation - Simplify First 오답률 100%

오답

Find $\dfrac{d y}{d x}$ and $\dfrac{d y}{d t}$. $y = \dfrac{t}{x^2} + \dfrac{x}{t}$

(미작성)

정답

$\dfrac{d y}{d x} = \dfrac{-2t}{x^3} + \dfrac{1}{t}$, $\dfrac{d y}{d t} = \dfrac{1}{x^2} - \dfrac{x}{t^2}$

위 정답과 비교하여 채점하세요:

해설 없음

27 Differentiation - Product Rule 오답률 100%

오답

Find the derivative of $f(x) = (1 + 2x^2)(x - x^2)$ in two ways: by using the Product Rule and by performing the multiplication first. Do your answers agree?

(미작성)

정답

$f'(x) = 1 + 2x - 3x^2 - 8x^3$

위 정답과 비교하여 채점하세요:

해설 없음

28 Differentiation - Quotient Rule 오답률 100%

오답

Find the derivative of $F(x) = \dfrac{x^4 - 5x^3 + \sqrt{x}}{x^2}$ in two ways: by using the Quotient Rule and by simplifying first. Do your answers agree?

(미작성)

정답

$F'(x) = 2x - 5 - \dfrac{3}{2} x^{\dfrac{-5}{2}}$

위 정답과 비교하여 채점하세요:

해설 없음

29 Differentiation - Product Rule 오답률 100%

오답

Use the Product Rule to find the derivative of the function. $f(x) = (3x^2 - 5x) x^2$

(미작성)

정답

$f'(x) = 12x^3 - 15x^2$

위 정답과 비교하여 채점하세요:

해설 없음

30 Differentiation - Product Rule 오답률 100%

오답

Use the Product Rule to find the derivative of the function. $y = (10x^2 + 7x - 2)(2 - x^2)$

(미작성)

정답

$y' = (20x + 7)(2 - x^2) + (10x^2 + 7x - 2)(-2x)$

위 정답과 비교하여 채점하세요:

해설 없음

31 Differentiation - Product Rule 오답률 100%

오답

Use the Product Rule to find the derivative of the function. $y = (4x^2 + 3)(2x + 5)$

(미작성)

정답

$y' = 8x(2x + 5) + (4x^2 + 3)(2) = 24x^2 + 40x + 6$

위 정답과 비교하여 채점하세요:

해설 없음

32 Differentiation - Product Rule 오답률 100%

오답

Use the Product Rule to find the derivative of the function. $g(x) = \sqrt{x}(x + 2 \sqrt{x})$

(미작성)

정답

$g'(x) = \dfrac{1}{2 \sqrt{x}}(x + 2 \sqrt{x}) + \sqrt{x}\left(1 + \dfrac{1}{\sqrt{x}}\right)$

위 정답과 비교하여 채점하세요:

해설 없음

33 Differentiation - Quotient Rule 오답률 100%

오답

Use the Quotient Rule to find the derivative of the function. $y = \dfrac{5x}{1 + x}$

(미작성)

정답

$y' = \dfrac{5(1 + x) - 5x}{(1 + x)^2} = \dfrac{5}{(1 + x)^2}$

위 정답과 비교하여 채점하세요:

해설 없음

34 Differentiation - Quotient Rule 오답률 100%

오답

Use the Quotient Rule to find the derivative of the function. $y = \dfrac{x^2}{1 - x}$

(미작성)

정답

$y' = \dfrac{2x(1 - x) + x^2}{(1 - x)^2} = \dfrac{2x - x^2}{(1 - x)^2}$

위 정답과 비교하여 채점하세요:

해설 없음

35 Differentiation - Quotient Rule 오답률 100%

오답

Use the Quotient Rule to find the derivative of the function. $g(t) = \dfrac{3 - 2t}{5t + 1}$

(미작성)

정답

$g'(t) = \dfrac{-2(5t + 1) - (3 - 2t)(5)}{(5t + 1)^2} = \dfrac{13}{(5t + 1)^2}$

위 정답과 비교하여 채점하세요:

해설 없음

36 Differentiation - Quotient Rule 오답률 100%

오답

Use the Quotient Rule to find the derivative of the function. $G(u) = \dfrac{6u^4 - 5u}{u + 1}$

(미작성)

정답

$G'(u) = \dfrac{(24u^3 - 5)(u + 1) - (6u^4 - 5u)}{(u + 1)^2}$

위 정답과 비교하여 채점하세요:

해설 없음

37 Differentiation - Mixed 오답률 100%

오답

Differentiate. $f(t) = \dfrac{5t}{t^3 - t - 1}$

(미작성)

정답

$f'(t) = \dfrac{5(t^3 - t - 1) - 5t(3t^2 - 1)}{(t^3 - t - 1)^2}$

위 정답과 비교하여 채점하세요:

해설 없음

38 Differentiation - Mixed 오답률 100%

오답

Differentiate. $F(x) = \dfrac{1}{2x^3 - 6x^2 + 5}$

(미작성)

정답

$F'(x) = \dfrac{-(6x^2 - 12x)}{(2x^3 - 6x^2 + 5)^2}$

위 정답과 비교하여 채점하세요:

해설 없음

39 Differentiation - Mixed 오답률 100%

오답

Differentiate. $y = \dfrac{s - \sqrt{s}}{s^2}$

(미작성)

정답

$y' = -s^{-2} + \dfrac{3}{2} s^{\dfrac{-5}{2}}$

위 정답과 비교하여 채점하세요:

해설 없음

40 Differentiation - Mixed 오답률 100%

오답

Differentiate. $y = \dfrac{\sqrt{x}}{\sqrt{x} + 1}$

(미작성)

정답

$y' = \dfrac{1}{2 \sqrt{x}(\sqrt{x} + 1)^2}$

위 정답과 비교하여 채점하세요:

해설 없음

41 Differentiation - Mixed 오답률 100%

오답

Differentiate. $F(x) = \dfrac{2x^5 + x^4 - 6x}{x^3}$

(미작성)

정답

$F'(x) = 2 + x^{-2} + 12x^{-3}$

위 정답과 비교하여 채점하세요:

해설 없음

42 Differentiation - Mixed 오답률 100%

오답

Differentiate. $y = \dfrac{(u + 2)^2}{1 - u}$

(미작성)

정답

$y' = \dfrac{2(u + 2)(1 - u) + (u + 2)^2}{(1 - u)^2}$

위 정답과 비교하여 채점하세요:

해설 없음

43 Differentiation - Mixed 오답률 100%

오답

Differentiate. $H(u) = (u - \sqrt{u})(u + \sqrt{u})$

(미작성)

정답

$H'(u) = 2u - 1$

위 정답과 비교하여 채점하세요:

해설 없음

44 Differentiation - Mixed 오답률 100%

오답

Differentiate. $A(v) = v^{\dfrac{2}{3}}(2v^2 + 1 - v^{-2})$

(미작성)

정답

$A'(v) = \dfrac{2}{3} v^{\dfrac{-1}{3}}(2v^2 + 1 - v^{-2}) + v^{\dfrac{2}{3}}(4v + 2v^{-3})$

위 정답과 비교하여 채점하세요:

해설 없음

45 Differentiation - Mixed 오답률 100%

오답

Differentiate. $J(u) = \left(\dfrac{1}{u} + \dfrac{1}{u^2}\right)\left(u + \dfrac{1}{u}\right)$

(미작성)

정답

$J'(u) = (-u^{-2} - 2u^{-3})(u + u^{-1}) + (u^{-1} + u^{-2})(1 - u^{-2})$

위 정답과 비교하여 채점하세요:

해설 없음

46 Differentiation - Mixed 오답률 100%

오답

Differentiate. $h(w) = (w^2 + 3w)(w^{-1} - w^{-4})$

(미작성)

정답

$h'(w) = (2w + 3)(w^{-1} - w^{-4}) + (w^2 + 3w)(-w^{-2} + 4w^{-5})$

위 정답과 비교하여 채점하세요:

해설 없음

47 Differentiation - Mixed 오답률 100%

오답

Differentiate. $f(t) = \dfrac{\sqrt[3]{t}}{t - 3}$

(미작성)

정답

$f'(t) = \dfrac{\dfrac{1}{3} t^{\dfrac{-2}{3}}(t - 3) - t^{\dfrac{1}{3}}}{(t - 3)^2}$

위 정답과 비교하여 채점하세요:

해설 없음

48 Differentiation - Mixed 오답률 100%

오답

Differentiate. $y = \dfrac{c x}{1 + c x}$

(미작성)

정답

$y' = \dfrac{c}{(1 + c x)^2}$

위 정답과 비교하여 채점하세요:

해설 없음

49 Differentiation - Mixed 오답률 100%

오답

Differentiate. $G(y) = \dfrac{B}{A y^3 + B}$

(미작성)

정답

$G'(y) = \dfrac{-3A B y^2}{(A y^3 + B)^2}$

위 정답과 비교하여 채점하세요:

해설 없음

50 Differentiation - Mixed 오답률 100%

오답

Differentiate. $F(t) = \dfrac{A t}{B t^2 + C t^3}$

(미작성)

정답

$F'(t) = \dfrac{A(B t^2 + C t^3) - A t(2B t + 3C t^2)}{(B t^2 + C t^3)^2}$

위 정답과 비교하여 채점하세요:

해설 없음

51 Differentiation - Mixed 오답률 100%

오답

Differentiate. $f(x) = \dfrac{x}{x + \dfrac{c}{x}}$

(미작성)

정답

$f'(x) = \dfrac{x^2 + c - x \cdot \dfrac{x^2 - c}{x^2}}{\left(x + \dfrac{c}{x}\right)^2}$

위 정답과 비교하여 채점하세요:

해설 없음

52 Differentiation - Mixed 오답률 100%

오답

Differentiate. $f(x) = \dfrac{a x + b}{c x + d}$

(미작성)

정답

$f'(x) = \dfrac{a d - b c}{(c x + d)^2}$

위 정답과 비교하여 채점하세요:

해설 없음

53 Differentiation - General Polynomial 오답률 100%

오답

The general polynomial of degree $n$ has the form $P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0$, where $a_n \neq 0$. Find $P'(x)$.

(미작성)

정답

$P'(x) = n a_n x^{n-1} + (n-1) a_{n-1} x^{n-2} + \cdots + 2a_2 x + a_1$

위 정답과 비교하여 채점하세요:

해설 없음

54 Differentiation - Graph Comparison 오답률 100%

오답

Find $f'(x)$. Compare the graphs of $f$ and $f'$ and use them to explain why your answer is reasonable. $f(x) = x^4 - 2x^3 + x^2$

(미작성)

정답

$f'(x) = 4x^3 - 6x^2 + 2x$

위 정답과 비교하여 채점하세요:

해설 없음

55 Differentiation - Graph Comparison 오답률 100%

오답

Find $f'(x)$. Compare the graphs of $f$ and $f'$ and use them to explain why your answer is reasonable. $f(x) = 3x^{15} - 5x^3 + 3$

(미작성)

정답

$f'(x) = 45x^{14} - 15x^2$

위 정답과 비교하여 채점하세요:

해설 없음

56 Differentiation - Graph Comparison 오답률 100%

오답

Find $f'(x)$. Compare the graphs of $f$ and $f'$ and use them to explain why your answer is reasonable. $f(x) = x + \dfrac{1}{x}$

(미작성)

정답

$f'(x) = 1 - \dfrac{1}{x^2}$

위 정답과 비교하여 채점하세요:

해설 없음

57 Differentiation - Graph Comparison 오답률 100%

오답

(a) Graph $f(x) = x^4 - 3x^3 - 6x^2 + 7x + 30$ in the viewing rectangle $[-3, 5]$ by $[-10, 50]$. (b) On a separate graph, sketch $f'$ by hand, using the graph in part (a) to estimate the slope of the tangent line at selected points. (c) Calculate $f'(x)$ and use this expression to graph $f'$. Compare with your sketch in part (b).

(미작성)

정답

$f'(x) = 4x^3 - 9x^2 - 12x + 7$

위 정답과 비교하여 채점하세요:

해설 없음

58 Differentiation - Graph Comparison 오답률 100%

오답

(a) Graph $g(x) = \dfrac{x^2}{x^2 + 1}$ in the viewing rectangle $[-4, 4]$ by $[-1, 1.5]$. (b) On a separate graph, sketch $g'$ by hand, using the graph in part (a) to estimate the slope of the tangent line at selected points. (c) Calculate $g'(x)$ and use this expression to graph $g'$. Compare with your sketch in part (b).

(미작성)

정답

$g'(x) = \dfrac{2x}{(x^2 + 1)^2}$

위 정답과 비교하여 채점하세요:

해설 없음

59 Differentiation - Tangent Line 오답률 100%

오답

Find an equation of the tangent line to the curve at the given point. $y = \dfrac{2x}{x + 1}$, $(1, 1)$

(미작성)

정답

$y - 1 = \dfrac{1}{2}(x - 1)$

위 정답과 비교하여 채점하세요:

해설 없음

60 Differentiation - Tangent Line 오답률 100%

오답

Find an equation of the tangent line to the curve at the given point. $y = 2x^3 - x^2 + 2$, $(1, 3)$

(미작성)

정답

$y - 3 = 4(x - 1)$

위 정답과 비교하여 채점하세요:

해설 없음

61 Differentiation - Tangent Line 오답률 100%

오답

Find equations of the tangent line and normal line to the curve at the given point. $y = x + \sqrt{x}$, $(1, 2)$

(미작성)

정답

Tangent: $y - 2 = \dfrac{3}{2}(x - 1)$; Normal: $y - 2 = \dfrac{-2}{3}(x - 1)$

위 정답과 비교하여 채점하세요:

해설 없음

62 Differentiation - Tangent Line 오답률 100%

오답

Find equations of the tangent line and normal line to the curve at the given point. $y = x^{\dfrac{3}{2}}$, $(1, 1)$

(미작성)

정답

Tangent: $y - 1 = \dfrac{3}{2}(x - 1)$; Normal: $y - 1 = \dfrac{-2}{3}(x - 1)$

위 정답과 비교하여 채점하세요:

해설 없음

63 Differentiation - Tangent Line 오답률 100%

오답

Find equations of the tangent line and normal line to the curve at the given point. $y = \dfrac{3x}{1 + 5x^2}$, $\left(1, \dfrac{1}{2}\right)$

(미작성)

정답

Tangent: $y - \dfrac{1}{2} = \dfrac{-7}{36}(x - 1)$; Normal: $y - \dfrac{1}{2} = \dfrac{36}{7}(x - 1)$

위 정답과 비교하여 채점하세요:

해설 없음

64 Differentiation - Tangent Line 오답률 100%

오답

Find equations of the tangent line and normal line to the curve at the given point. $y = \dfrac{\sqrt{x}}{x + 1}$, $(4, 0.4)$

(미작성)

정답

Tangent: $y - 0.4 = \dfrac{-3}{100}(x - 4)$; Normal: $y - 0.4 = \dfrac{100}{3}(x - 4)$

위 정답과 비교하여 채점하세요:

해설 없음

65 Differentiation - Tangent Line 오답률 100%

오답

(a) The curve $y = \dfrac{1}{1 + x^2}$ is called a witch of Maria Agnesi. Find an equation of the tangent line to this curve at the point $\left(-1, \dfrac{1}{2}\right)$. (b) Illustrate part (a) by graphing the curve and tangent line on the same screen.

(미작성)

정답

$y - \dfrac{1}{2} = \dfrac{1}{2}(x + 1)$

위 정답과 비교하여 채점하세요:

해설 없음

66 Differentiation - Tangent Line 오답률 100%

오답

(a) The curve $y = \dfrac{x}{1 + x^2}$ is called a serpentine. Find an equation of the tangent line to this curve at the point $(3, 0.3)$. (b) Illustrate part (a) by graphing the curve and tangent line on the same screen.

(미작성)

정답

$y - 0.3 = \dfrac{-8}{100}(x - 3)$

위 정답과 비교하여 채점하세요:

해설 없음

67 Differentiation - Higher Derivatives 오답률 100%

오답

Find the first and second derivatives of the function. $f(x) = 0.001x^5 - 0.02x^3$

(미작성)

정답

$f'(x) = 0.005x^4 - 0.06x^2$, $f''(x) = 0.02x^3 - 0.12x$

위 정답과 비교하여 채점하세요:

해설 없음

68 Differentiation - Higher Derivatives 오답률 100%

오답

Find the first and second derivatives of the function. $G(r) = \sqrt{r} + \sqrt[3]{r}$

(미작성)

정답

$G'(r) = \dfrac{1}{2} r^{\dfrac{-1}{2}} + \dfrac{1}{3} r^{\dfrac{-2}{3}}$, $G''(r) = -\dfrac{1}{4} r^{\dfrac{-3}{2}} - \dfrac{2}{9} r^{\dfrac{-5}{3}}$

위 정답과 비교하여 채점하세요:

해설 없음

69 Differentiation - Higher Derivatives 오답률 100%

오답

Find the first and second derivatives of the function. $f(x) = \dfrac{x^2}{1 + 2x}$

(미작성)

정답

$f'(x) = \dfrac{2x(1 + 2x) - 2x^2}{(1 + 2x)^2} = \dfrac{2x + 2x^2}{(1 + 2x)^2}$

위 정답과 비교하여 채점하세요:

해설 없음

70 Differentiation - Higher Derivatives 오답률 100%

오답

Find the first and second derivatives of the function. $f(x) = \dfrac{1}{3 - x}$

(미작성)

정답

$f'(x) = \dfrac{1}{(3 - x)^2}$, $f''(x) = \dfrac{2}{(3 - x)^3}$

위 정답과 비교하여 채점하세요:

해설 없음

71 Differentiation - Higher Derivatives 오답률 100%

오답

Find the first and second derivatives of the function. Check to see that your answers are reasonable by comparing the graphs of $f$, $f'$, and $f''$. $f(x) = 2x - 5x^{\dfrac{3}{4}}$

(미작성)

정답

$f'(x) = 2 - \dfrac{15}{4} x^{\dfrac{-1}{4}}$, $f''(x) = \dfrac{15}{16} x^{\dfrac{-5}{4}}$

위 정답과 비교하여 채점하세요:

해설 없음

72 Differentiation - Higher Derivatives 오답률 100%

오답

Find the first and second derivatives of the function. Check to see that your answers are reasonable by comparing the graphs of $f$, $f'$, and $f''$. $f(x) = \dfrac{x^2 - 1}{x^2 + 1}$

(미작성)

정답

$f'(x) = \dfrac{4x}{(x^2 + 1)^2}$

위 정답과 비교하여 채점하세요:

해설 없음

73 Differentiation - Motion 오답률 100%

오답

The equation of motion of a particle is $s = t^3 - 3t$, where $s$ is measured in meters and $t$ in seconds. Find (a) the velocity and acceleration as functions of $t$, (b) the acceleration after 2 s, and (c) the acceleration when the velocity is 0.

(미작성)

정답

(a) $v(t) = 3t^2 - 3$, $a(t) = 6t$; (b) $a(2) = 12$ m/s^2; (c) $v = 0$ when $t = 1$, $a(1) = 6$ m/s^2

위 정답과 비교하여 채점하세요:

해설 없음

74 Differentiation - Motion 오답률 100%

오답

The equation of motion of a particle is $s = t^4 - 2t^3 + t^2 - t$, where $s$ is in meters and $t$ is in seconds. (a) Find the velocity and acceleration as functions of $t$. (b) Find the acceleration after 1 s.

(미작성)

정답

(a) $v(t) = 4t^3 - 6t^2 + 2t - 1$, $a(t) = 12t^2 - 12t + 2$; (b) $a(1) = 2$ m/s^2

위 정답과 비교하여 채점하세요:

해설 없음

75 Differentiation - Applied 오답률 100%

오답

Biologists have proposed a cubic polynomial to model the length $L$ of Alaskan rockfish at age $A$: $L = 0.0155 A^3 - 0.372 A^2 + 3.95 A + 1.21$ where $L$ is measured in inches and $A$ in years. Calculate $\dfrac{d L}{d A} bar.v_{A = 12}$ and interpret the result.

(미작성)

정답

$\dfrac{d L}{d A} = 0.0465 A^2 - 0.744 A + 3.95$; at $A = 12$: $\dfrac{d L}{d A} bar.v_{A=12} \approx 0.742$ inches per year

위 정답과 비교하여 채점하세요:

해설 없음

76 Differentiation - Applied 오답률 100%

오답

The number of tree species $S$ in a given area $A$ in the Pasoh Forest Reserve in Malaysia has been modeled by the power function $S(A) = 0.882 A^{0.842}$. Find $S'(100)$ and interpret your answer.

(미작성)

정답

$S'(A) = 0.882 \cdot 0.842 A^{-0.158} = 0.742644 A^{-0.158}$; $S'(100) \approx 0.36$

위 정답과 비교하여 채점하세요:

해설 없음

77 Differentiation - Applied 오답률 100%

오답

According to Boyle's Law, when a sample of gas is compressed at a constant temperature, the pressure $P$ of the gas is inversely proportional to the volume $V$ of the gas. (a) Suppose that the pressure of a sample of air that occupies $0.106$ m^3 at $25^{\circ}$C is 50 kPa. Write $V$ as a function of $P$. (b) Calculate $\dfrac{d V}{d P}$ when $P = 50$ kPa. What is the meaning of the derivative? What are its units?

(미작성)

정답

(a) $V = \dfrac{5.3}{P}$; (b) $\dfrac{d V}{d P} = \dfrac{-5.3}{P^2}$; at $P = 50$: $\dfrac{d V}{d P} = -0.00212$ m^3 per kPa

위 정답과 비교하여 채점하세요:

해설 없음

78 Differentiation - Applied 오답률 100%

오답

The table shows data from an experiment in which the weights on a tire were varied and the tire life was measured. (a) Find a quadratic model for the data. (b) Use the model to estimate $\dfrac{d L}{d P}$ when $P = 30$ and when $P = 40$. Interpret the results.

(미작성)

정답

Answers depend on the quadratic regression model fitted to the data.

위 정답과 비교하여 채점하세요:

해설 없음

79 Differentiation - Given Values 오답률 100%

오답

If $f(5) = 1$, $f'(5) = 6$, $g(5) = -3$, $g'(5) = 2$, find the following numbers. (a) $(f g)'(5)$ (b) $\left(\dfrac{f}{g}\right)'(5)$ (c) $\left(\dfrac{g}{f}\right)'(5)$

(미작성)

정답

(a) $(f g)'(5) = f'(5) g(5) + f(5) g'(5) = 6(-3) + 1(2) = -16$; (b) $\left(\dfrac{f}{g}\right)'(5) = \dfrac{f'(5) g(5) - f(5) g'(5)}{[g(5)]^2} = \dfrac{-20}{9}$; (c) $\left(\dfrac{g}{f}\right)'(5) = \dfrac{g'(5) f(5) - g(5) f'(5)}{[f(5)]^2} = 20$

위 정답과 비교하여 채점하세요:

해설 없음

80 Differentiation - Given Values 오답률 100%

오답

If $f(4) = 2$, $g(4) = 5$, $f'(4) = 6$, $g'(4) = -3$, find $h'(4)$ for each of the following. (a) $h(x) = 3f(x) + 8g(x)$ (b) $h(x) = f(x) g(x)$ (c) $h(x) = \dfrac{f(x)}{g(x)}$ (d) $h(x) = \dfrac{g(x)}{f(x) + g(x)}$

(미작성)

정답

(a) $h'(4) = 3(6) + 8(-3) = -6$; (b) $h'(4) = 6(5) + 2(-3) = 24$; (c) $h'(4) = \dfrac{6(5) - 2(-3)}{25} = \dfrac{36}{25}$; (d) $h'(4) = \dfrac{-3(2+5) - 5(6+(-3))}{(2+5)^2} = \dfrac{-36}{49}$

위 정답과 비교하여 채점하세요:

해설 없음

81 Differentiation - Given Values 오답률 100%

오답

If $f(x) = \sqrt{x} \cdot g(x)$, where $g(4) = 8$ and $g'(4) = 7$, find $f'(4)$.

(미작성)

정답

$f'(4) = \dfrac{1}{2 \sqrt{4}} g(4) + \sqrt{4} g'(4) = \dfrac{8}{4} + 2(7) = 2 + 14 = 16$

위 정답과 비교하여 채점하세요:

해설 없음

82 Differentiation - Given Values 오답률 100%

오답

If $h(2) = 4$ and $h'(2) = -3$, find $\dfrac{d}{d x}(\dfrac{h(x)}{x}) bar.v_{x = 2}$.

(미작성)

정답

$\dfrac{d}{d x}(\dfrac{h(x)}{x}) = \dfrac{h'(x) x - h(x)}{x^2}$; at $x = 2$: $\dfrac{-3(2) - 4}{4} = \dfrac{-10}{4} = \dfrac{-5}{2}$

위 정답과 비교하여 채점하세요:

해설 없음

83 Differentiation - Given Values 오답률 100%

오답

If $f$ and $g$ are the functions whose graphs are shown, let $u(x) = f(x) g(x)$ and $v(x) = \dfrac{f(x)}{g(x)}$. (a) Find $u'(1)$. (b) Find $v'(4)$.

(미작성)

정답

Read values from graph to apply product and quotient rules.

위 정답과 비교하여 채점하세요:

해설 없음

84 Differentiation - Given Values 오답률 100%

오답

Let $P(x) = F(x) G(x)$ and $Q(x) = \dfrac{F(x)}{G(x)}$, where $F$ and $G$ are the functions whose graphs are shown. (a) Find $P'(2)$. (b) Find $Q'(7)$.

(미작성)

정답

Read values from graph to apply product and quotient rules.

위 정답과 비교하여 채점하세요:

해설 없음

85 Differentiation - Abstract/General 오답률 100%

오답

If $g$ is a differentiable function, find an expression for the derivative of each of the following. (a) $y = x g(x)$ (b) $y = \dfrac{x}{g(x)}$ (c) $y = \dfrac{g(x)}{x}$

(미작성)

정답

(a) $y' = g(x) + x g'(x)$; (b) $y' = \dfrac{g(x) - x g'(x)}{[g(x)]^2}$; (c) $y' = \dfrac{x g'(x) - g(x)}{x^2}$

위 정답과 비교하여 채점하세요:

해설 없음

86 Differentiation - Abstract/General 오답률 100%

오답

If $f$ is a differentiable function, find an expression for the derivative of each of the following. (a) $y = x^2 f(x)$ (b) $y = \dfrac{f(x)}{x^2}$ (c) $y = \dfrac{x^2}{f(x)}$ (d) $y = \dfrac{1 + x f(x)}{\sqrt{x}}$

(미작성)

정답

(a) $y' = 2x f(x) + x^2 f'(x)$; (b) $y' = \dfrac{f'(x) x^2 - 2x f(x)}{x^4}$; (c) $y' = \dfrac{2x f(x) - x^2 f'(x)}{[f(x)]^2}$; (d) $y' = \dfrac{f(x) + x f'(x)}{\sqrt{x}} - \dfrac{1 + x f(x)}{2x^{\dfrac{3}{2}}}$

위 정답과 비교하여 채점하세요:

해설 없음

87 Differentiation - Finding Points 오답률 100%

오답

Find the points on the curve $y = x^3 + 3x^2 - 9x + 10$ where the tangent is horizontal.

(미작성)

정답

$y' = 3x^2 + 6x - 9 = 3(x + 3)(x - 1) = 0$ gives $x = -3$ and $x = 1$. Points: $(-3, 37)$ and $(1, 5)$.

위 정답과 비교하여 채점하세요:

해설 없음

88 Differentiation - Finding Points 오답률 100%

오답

For what values of $x$ does the graph of $f(x) = x^3 + 3x^2 + x + 3$ have a horizontal tangent?

(미작성)

정답

$f'(x) = 3x^2 + 6x + 1 = 0$ gives $x = \dfrac{-6 \pm \sqrt{24}}{6} = \dfrac{-3 \pm \sqrt{6}}{3}$

위 정답과 비교하여 채점하세요:

해설 없음

89 Differentiation - Finding Points 오답률 100%

오답

Show that the curve $y = 6x^3 + 5x - 3$ has no tangent line with slope 4.

(미작성)

정답

$y' = 18x^2 + 5 \geq 5 > 4$ for all $x$. Since $y' > 4$ always, no tangent line has slope 4.

위 정답과 비교하여 채점하세요:

해설 없음

90 Differentiation - Finding Points 오답률 100%

오답

Find an equation of the tangent line to the curve $y = x^4 + 1$ that is parallel to the line $32x - y = 15$.

(미작성)

정답

Slope = 32. $y' = 4x^3 = 32$ gives $x = 2$. Point: $(2, 17)$. Tangent: $y - 17 = 32(x - 2)$.

위 정답과 비교하여 채점하세요:

해설 없음

91 Differentiation - Finding Points 오답률 100%

오답

Find equations of both lines that are tangent to the curve $y = x^3 - 3x^2 + 3x - 3$ and are parallel to the line $3x - y = 15$.

(미작성)

정답

Slope = 3. $y' = 3x^2 - 6x + 3 = 3(x - 1)^2 = 3$ gives $x = 0$ or $x = 2$. Two tangent lines.

위 정답과 비교하여 채점하세요:

해설 없음

92 Differentiation - Finding Points 오답률 100%

오답

Find equations of the tangent lines to the curve $y = \dfrac{x - 1}{x + 1}$ that are parallel to the line $x - 2y = 2$.

(미작성)

정답

Slope = $\dfrac{1}{2}$. $y' = \dfrac{2}{(x + 1)^2} = \dfrac{1}{2}$ gives $(x + 1)^2 = 4$, so $x = 1$ or $x = -3$.

위 정답과 비교하여 채점하세요:

해설 없음

93 Differentiation - Finding Points 오답률 100%

오답

Find an equation of the normal line to the parabola $y = \sqrt{x}$ that is parallel to the line $2x + y = 1$.

(미작성)

정답

Normal slope = $-2$. Tangent slope = $\dfrac{1}{2}$. $y' = \dfrac{1}{2 \sqrt{x}} = \dfrac{1}{2}$ gives $x = 1$, $y = 1$. Normal: $y - 1 = -2(x - 1)$.

위 정답과 비교하여 채점하세요:

해설 없음

94 Differentiation - Finding Points 오답률 100%

오답

Where does the normal line to the parabola $y = x^2 - 1$ at the point $(-1, 0)$ intersect the parabola a second time? Illustrate with a sketch.

(미작성)

정답

Tangent slope at $(-1, 0)$: $y' = 2x$, so slope = $-2$. Normal slope = $\dfrac{1}{2}$. Normal: $y = \dfrac{1}{2}(x + 1)$. Solve with $y = x^2 - 1$.

위 정답과 비교하여 채점하세요:

해설 없음

95 Differentiation - Finding Points 오답률 100%

오답

Draw a diagram to show that there are two tangent lines to the parabola $y = x^2$ that pass through the point $(0, -4)$. Find the coordinates of the points where these tangent lines touch the parabola.

(미작성)

정답

Tangent at $(a, a^2)$: $y - a^2 = 2a(x - a)$. Passes through $(0, -4)$: $-4 - a^2 = -2a^2$, so $a^2 = 4$, $a = \pm 2$. Points: $(2, 4)$ and $(-2, 4)$.

위 정답과 비교하여 채점하세요:

해설 없음

96 Differentiation - Finding Points 오답률 100%

오답

(a) Find equations of both lines through the point $(2, -3)$ that are tangent to the parabola $y = x^2 + x$. (b) Show that there is no line through the point $(2, 7)$ that is tangent to the parabola. Then draw a diagram to see why.

(미작성)

정답

(a) Tangent at $(a, a^2 + a)$ with slope $2a + 1$ passes through $(2, -3)$. Solve to find $a$.

위 정답과 비교하여 채점하세요:

해설 없음

97 Differentiation - Finding Points 오답률 100%

오답

For what values of $a$ and $b$ is the line $2x + y = b$ tangent to the parabola $y = a x^2$ when $x = 2$?

(미작성)

정답

At $x = 2$: slope of tangent = $2a(2) = 4a = -2$, so $a = \dfrac{-1}{2}$. Then $y = \dfrac{-1}{2}(4) = -2$, and $b = 2(2) + (-2) = 2$.

위 정답과 비교하여 채점하세요:

해설 없음

98 Differentiation - Patterns/nth 오답률 100%

오답

Find the $n$th derivative of each function by calculating the first few derivatives and observing the pattern that occurs. (a) $f(x) = x^n$ (b) $f(x) = \dfrac{1}{x}$

(미작성)

정답

(a) $f^{(n)}(x) = n!$; (b) $f^{(n)}(x) = \dfrac{(-1)^n n!}{x^{n+1}}$

위 정답과 비교하여 채점하세요:

해설 없음

99 Differentiation - Patterns/nth 오답률 100%

오답

Find a second-degree polynomial $P$ such that $P(2) = 5$, $P'(2) = 3$, and $P''(2) = 2$.

(미작성)

정답

Let $P(x) = a x^2 + b x + c$. $P''(x) = 2a = 2$, so $a = 1$. $P'(x) = 2x + b$, $P'(2) = 4 + b = 3$, $b = -1$. $P(2) = 4 - 2 + c = 5$, $c = 3$. $P(x) = x^2 - x + 3$.

위 정답과 비교하여 채점하세요:

해설 없음

100 Differentiation - Proof/Theory 오답률 100%

오답

The equation $y'' + y' - 2y = x^2$ is called a differential equation because it involves an unknown function $y$ and its derivatives $y'$ and $y''$. Find constants $A$, $B$, and $C$ such that the function $y = A x^2 + B x + C$ satisfies this equation. (Differential equations will be studied in detail in Chapter 7.)

(미작성)

정답

$y' = 2A x + B$, $y'' = 2A$. Substituting: $2A + 2A x + B - 2(A x^2 + B x + C) = x^2$. Matching: $-2A = 1$, $2A - 2B = 0$, $2A + B - 2C = 0$. So $A = \dfrac{-1}{2}$, $B = \dfrac{-1}{2}$, $C = \dfrac{-3}{4}$.

위 정답과 비교하여 채점하세요:

해설 없음

101 Differentiation - Proof/Theory 오답률 100%

오답

Find a cubic function $y = a x^3 + b x^2 + c x + d$ whose graph has horizontal tangents at the points $(-2, 6)$ and $(2, 0)$.

(미작성)

정답

$y' = 3a x^2 + 2b x + c$. Conditions: $y'(-2) = 0$, $y'(2) = 0$, $y(-2) = 6$, $y(2) = 0$. Solve the system.

위 정답과 비교하여 채점하세요:

해설 없음

102 Differentiation - Proof/Theory 오답률 100%

오답

Find a parabola $y = a x^2 + b x + c$ that has slope 4 at $x = 1$, slope $-8$ at $x = -1$, and passes through the point $(2, 15)$.

(미작성)

정답

$y' = 2a x + b$. $y'(1) = 2a + b = 4$, $y'(-1) = -2a + b = -8$. Solving: $a = 3$, $b = -2$. $y(2) = 12 - 4 + c = 15$, $c = 7$. $y = 3x^2 - 2x + 7$.

위 정답과 비교하여 채점하세요:

해설 없음

103 Differentiation - Applied 오답률 100%

오답

In 2018 the population of Boulder, Colorado, was 108,250 and was increasing at a rate of about 1300 people per year. The average annual income was \$62,370 per capita, and this average was increasing at about \$2500 per year. Use the Product Rule to estimate the rate at which total personal income was rising in Boulder in 2018. Explain the meaning of each term in the Product Rule.

(미작성)

정답

Total income $= P \cdot I$. Rate $= P' \cdot I + P \cdot I' = 1300 \cdot 62370 + 108250 \cdot 2500 \approx \$351,706,000$ per year.

위 정답과 비교하여 채점하세요:

해설 없음

104 Differentiation - Applied 오답률 100%

오답

A manufacturer of fabric produces rolls of material and the total revenue is $R(p) = p f(p)$ dollars, where $p$ is the price per yard and $f(p)$ is the number of yards sold. (a) What does it mean to say that $f(20) = 10000$ and $f'(20) = -350$? (b) Find $R'(20)$ and interpret your answer.

(미작성)

정답

(a) At \$20 per yard, 10000 yards are sold; increasing price by \$1 decreases demand by 350 yards. (b) $R'(20) = f(20) + 20 f'(20) = 10000 + 20(-350) = 3000$ dollars per dollar.

위 정답과 비교하여 채점하세요:

해설 없음

105 Differentiation - Applied 오답률 100%

오답

The Michaelis-Menten equation for the enzyme chymotrypsin is $v = \dfrac{0.14 [S]}{0.015 + [S]}$, where $v$ is the rate of an enzymatic reaction and $[S]$ is the concentration of a substrate $S$. Calculate $\dfrac{d v}{d [S]}$ and interpret the result.

(미작성)

정답

$\dfrac{d v}{d [S]} = \dfrac{0.14 \cdot 0.015}{(0.015 + [S])^2} = \dfrac{0.0021}{(0.015 + [S])^2}$

위 정답과 비교하여 채점하세요:

해설 없음

106 Differentiation - Applied 오답률 100%

오답

The biomass of a guppy population in a small aquarium is modeled using the Product Rule. Let $N(t)$ be the number of guppies and $w(t)$ be the average weight of each guppy at time $t$, and $B(t) = N(t) w(t)$ is the total biomass. Use the Product Rule to find $B'(t)$ and interpret each term.

(미작성)

정답

$B'(t) = N'(t) w(t) + N(t) w'(t)$. First term: biomass increase from population growth. Second term: biomass increase from individual weight gain.

위 정답과 비교하여 채점하세요:

해설 없음

107 Differentiation - Proof/Theory 오답률 100%

오답

(a) If $F(x) = f(x) g(x) h(x)$ and $F'$, $f'$, $g'$, and $h'$ all exist, show that $F'(x) = f'(x) g(x) h(x) + f(x) g'(x) h(x) + f(x) g(x) h'(x)$. (b) By taking $f = g = h$ in part (a), show that $\dfrac{d}{d x} [f(x)]^3 = 3 [f(x)]^2 f'(x)$. (c) Use part (b) to differentiate $y = (x^4 + 3x^3 + 17x + 82)^3$.

(미작성)

정답

(a) Apply Product Rule to $F = (f g) h$. (b) Set $f = g = h$. (c) $y' = 3(x^4 + 3x^3 + 17x + 82)^2(4x^3 + 9x^2 + 17)$.

위 정답과 비교하여 채점하세요:

해설 없음

108 Differentiation - Proof/Theory 오답률 100%

오답

(a) Use the Quotient Rule to prove the Reciprocal Rule: if $g$ is differentiable, then $\dfrac{d}{d x} [\dfrac{1}{g(x)}] = \dfrac{-g'(x)}{[g(x)]^2}$. (b) Use the Reciprocal Rule to differentiate the function in Exercise 38.

(미작성)

정답

(a) Apply Quotient Rule with $f(x) = 1$. (b) $F'(x) = \dfrac{-(6x^2 - 12x)}{(2x^3 - 6x^2 + 5)^2}$.

위 정답과 비교하여 채점하세요:

해설 없음

109 Differentiation - Proof/Theory 오답률 100%

오답

Use the Product Rule to prove the Quotient Rule. [Hint: Write $f(x) = [\dfrac{f(x)}{g(x)}] \cdot g(x)$.]

(미작성)

정답

Let $F(x) = \dfrac{f(x)}{g(x)}$, so $f(x) = F(x) g(x)$. By the Product Rule, $f'(x) = F'(x) g(x) + F(x) g'(x)$. Solve for $F'(x)$.

위 정답과 비교하여 채점하세요:

해설 없음

110 Differentiation - Proof/Theory 오답률 100%

오답

If $F(x) = f(x) g(x)$, where $f$ and $g$ have derivatives of all orders, show that: (a) $F'' = f'' g + 2 f' g' + f g''$ (b) $F''' = f''' g + 3 f'' g' + 3 f' g'' + f g'''$ (c) Find a similar formula for $F^{(4)}$. (d) Guess a formula for $F^{(n)}$.

(미작성)

정답

(a) Differentiate $F' = f' g + f g'$ again. (b) Differentiate $F''$. (c) $F^{(4)} = f^{(4)} g + 4 f''' g' + 6 f'' g'' + 4 f' g''' + f g^{(4)}$. (d) $F^{(n)} = \displaystyle\sum_{k=0}^n \binom{n}{k} f^{(n-k)} g^{(k)}$ (Leibniz rule).

위 정답과 비교하여 채점하세요:

해설 없음

문제별 결과

다시 풀기