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Stewart 9th Section 2.3: Differentiation Formulas
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1
Differentiation - Basic
Error Rate 100%
Wrong
\( g(x) = 4x + 7 \)
(No answer submitted)
Answer
\(g'(x) = 4\)
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No explanation
2
Differentiation - Basic
Error Rate 100%
Wrong
\( g(t) = 5t + 4t^2 \)
(No answer submitted)
Answer
\(g'(t) = 5 + 8t\)
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No explanation
3
Differentiation - Basic
Error Rate 100%
Wrong
\( f(x) = x^{75} - x + 3 \)
(No answer submitted)
Answer
\(f'(x) = 75x^{74} - 1\)
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No explanation
4
Differentiation - Basic
Error Rate 100%
Wrong
\( g(x) = \dfrac{7}{4} x^2 - 3x + 12 \)
(No answer submitted)
Answer
\(g'(x) = \dfrac{7}{2} x - 3\)
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No explanation
5
Differentiation - Basic
Error Rate 100%
Wrong
\( W(v) = 1.8 v^{-3} \)
(No answer submitted)
Answer
\(W'(v) = -5.4 v^{-4}\)
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No explanation
6
Differentiation - Basic
Error Rate 100%
Wrong
\( r(z) = z^{-5} - z^{\dfrac{1}{2}} \)
(No answer submitted)
Answer
\(r'(z) = -5z^{-6} - \dfrac{1}{2} z^{\dfrac{-1}{2}}\)
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No explanation
7
Differentiation - Basic
Error Rate 100%
Wrong
\( f(x) = x^{\dfrac{3}{2}} + x^{-3} \)
(No answer submitted)
Answer
\(f'(x) = \dfrac{3}{2} x^{\dfrac{1}{2}} - 3x^{-4}\)
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No explanation
8
Differentiation - Basic
Error Rate 100%
Wrong
\( V(t) = t^{\dfrac{-3}{5}} + t^4 \)
(No answer submitted)
Answer
\(V'(t) = \dfrac{-3}{5} t^{\dfrac{-8}{5}} + 4t^3\)
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No explanation
9
Differentiation - Basic
Error Rate 100%
Wrong
\( s(t) = \dfrac{1}{t} + \dfrac{1}{t^2} \)
(No answer submitted)
Answer
\(s'(t) = -\dfrac{1}{t^2} - \dfrac{2}{t^3}\)
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No explanation
10
Differentiation - Basic
Error Rate 100%
Wrong
\( r(t) = \dfrac{a}{t^2} + \dfrac{b}{t^4} \)
(No answer submitted)
Answer
\(r'(t) = \dfrac{-2a}{t^3} + \dfrac{-4b}{t^5}\)
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No explanation
11
Differentiation - Basic
Error Rate 100%
Wrong
\( y = 2x + \sqrt{x} \)
(No answer submitted)
Answer
\(y' = 2 + \dfrac{1}{2 \sqrt{x}}\)
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No explanation
12
Differentiation - Basic
Error Rate 100%
Wrong
\( h(w) = \sqrt{2} w - \sqrt{2} \)
(No answer submitted)
Answer
\(h'(w) = \sqrt{2}\)
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No explanation
13
Differentiation - Basic
Error Rate 100%
Wrong
\( g(x) = \dfrac{1}{\sqrt{x}} + \sqrt[5]{x} \)
(No answer submitted)
Answer
\(g'(x) = -\dfrac{1}{2} x^{\dfrac{-3}{2}} + \dfrac{1}{5} x^{\dfrac{-4}{5}}\)
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No explanation
14
Differentiation - Basic
Error Rate 100%
Wrong
\( S(R) = 4 \pi R^2 \)
(No answer submitted)
Answer
\(S'(R) = 8 \pi R\)
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No explanation
15
Differentiation - Simplify First
Error Rate 100%
Wrong
\( f(x) = x^3 (x + 3) \)
(No answer submitted)
Answer
\(f'(x) = 4x^3 + 9x^2\)
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No explanation
16
Differentiation - Simplify First
Error Rate 100%
Wrong
\( F(t) = (2t - 3)^2 \)
(No answer submitted)
Answer
\(F'(t) = 8t - 12\)
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No explanation
17
Differentiation - Simplify First
Error Rate 100%
Wrong
\( f(x) = \dfrac{3x^2 + x^3}{x} \)
(No answer submitted)
Answer
\(f'(x) = 3 + 2x\)
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No explanation
18
Differentiation - Simplify First
Error Rate 100%
Wrong
\( y = \dfrac{\sqrt{x} + x}{x^2} \)
(No answer submitted)
Answer
\(y' = -\dfrac{3}{2} x^{\dfrac{-5}{2}} - x^{-2}\)
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No explanation
19
Differentiation - Simplify First
Error Rate 100%
Wrong
\( G(q) = (1 + q^{-1})^2 \)
(No answer submitted)
Answer
\(G'(q) = -2q^{-2} - 2q^{-3}\)
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No explanation
20
Differentiation - Simplify First
Error Rate 100%
Wrong
\( G(t) = \sqrt{5t} + \dfrac{\sqrt{7}}{t} \)
(No answer submitted)
Answer
\(G'(t) = \dfrac{\sqrt{5}}{2 \sqrt{t}} - \dfrac{\sqrt{7}}{t^2}\)
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No explanation
21
Differentiation - Simplify First
Error Rate 100%
Wrong
\( G(r) = \dfrac{3r^{\dfrac{3}{2}} + r^{\dfrac{5}{2}}}{r} \)
(No answer submitted)
Answer
\(G'(r) = \dfrac{3}{2} r^{\dfrac{-1}{2}} + \dfrac{3}{2} r^{\dfrac{1}{2}}\)
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No explanation
22
Differentiation - Simplify First
Error Rate 100%
Wrong
\( F(z) = \dfrac{A + B z + C z^2}{z^2} \)
(No answer submitted)
Answer
\(F'(z) = -2A z^{-3} - B z^{-2}\)
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No explanation
23
Differentiation - Simplify First
Error Rate 100%
Wrong
\( P(w) = \dfrac{2w^2 - w + 4}{\sqrt{w}} \)
(No answer submitted)
Answer
\(P'(w) = 3w^{\dfrac{1}{2}} - \dfrac{1}{2} w^{\dfrac{-1}{2}} - 2w^{\dfrac{-3}{2}}\)
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No explanation
24
Differentiation - Simplify First
Error Rate 100%
Wrong
\( D(t) = \dfrac{1 + 16t^2}{(4t)^3} \)
(No answer submitted)
Answer
\(D'(t) = -\dfrac{3}{64} t^{-4} + \dfrac{1}{4} t^{-2}\)
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No explanation
25
Differentiation - Simplify First
Error Rate 100%
Wrong
Find \(\dfrac{d y}{d x}\) and \(\dfrac{d y}{d t}\).
\(y = t x^2 + t^3 x\)
(No answer submitted)
Answer
\(\dfrac{d y}{d x} = 2t x + t^3\), \(\dfrac{d y}{d t} = x^2 + 3t^2 x\)
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No explanation
26
Differentiation - Simplify First
Error Rate 100%
Wrong
Find \(\dfrac{d y}{d x}\) and \(\dfrac{d y}{d t}\).
\(y = \dfrac{t}{x^2} + \dfrac{x}{t}\)
(No answer submitted)
Answer
\(\dfrac{d y}{d x} = \dfrac{-2t}{x^3} + \dfrac{1}{t}\), \(\dfrac{d y}{d t} = \dfrac{1}{x^2} - \dfrac{x}{t^2}\)
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No explanation
27
Differentiation - Product Rule
Error Rate 100%
Wrong
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?
(No answer submitted)
Answer
\(f'(x) = 1 + 2x - 3x^2 - 8x^3\)
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No explanation
28
Differentiation - Quotient Rule
Error Rate 100%
Wrong
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?
(No answer submitted)
Answer
\(F'(x) = 2x - 5 - \dfrac{3}{2} x^{\dfrac{-5}{2}}\)
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No explanation
29
Differentiation - Product Rule
Error Rate 100%
Wrong
Use the Product Rule to find the derivative of the function.
\(f(x) = (3x^2 - 5x) x^2\)
(No answer submitted)
Answer
\(f'(x) = 12x^3 - 15x^2\)
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No explanation
30
Differentiation - Product Rule
Error Rate 100%
Wrong
Use the Product Rule to find the derivative of the function.
\(y = (10x^2 + 7x - 2)(2 - x^2)\)
(No answer submitted)
Answer
\(y' = (20x + 7)(2 - x^2) + (10x^2 + 7x - 2)(-2x)\)
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No explanation
31
Differentiation - Product Rule
Error Rate 100%
Wrong
Use the Product Rule to find the derivative of the function.
\(y = (4x^2 + 3)(2x + 5)\)
(No answer submitted)
Answer
\(y' = 8x(2x + 5) + (4x^2 + 3)(2) = 24x^2 + 40x + 6\)
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No explanation
32
Differentiation - Product Rule
Error Rate 100%
Wrong
Use the Product Rule to find the derivative of the function.
\(g(x) = \sqrt{x}(x + 2 \sqrt{x})\)
(No answer submitted)
Answer
\(g'(x) = \dfrac{1}{2 \sqrt{x}}(x + 2 \sqrt{x}) + \sqrt{x}\left(1 + \dfrac{1}{\sqrt{x}}\right)\)
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No explanation
33
Differentiation - Quotient Rule
Error Rate 100%
Wrong
Use the Quotient Rule to find the derivative of the function.
\(y = \dfrac{5x}{1 + x}\)
(No answer submitted)
Answer
\(y' = \dfrac{5(1 + x) - 5x}{(1 + x)^2} = \dfrac{5}{(1 + x)^2}\)
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No explanation
34
Differentiation - Quotient Rule
Error Rate 100%
Wrong
Use the Quotient Rule to find the derivative of the function.
\(y = \dfrac{x^2}{1 - x}\)
(No answer submitted)
Answer
\(y' = \dfrac{2x(1 - x) + x^2}{(1 - x)^2} = \dfrac{2x - x^2}{(1 - x)^2}\)
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No explanation
35
Differentiation - Quotient Rule
Error Rate 100%
Wrong
Use the Quotient Rule to find the derivative of the function.
\(g(t) = \dfrac{3 - 2t}{5t + 1}\)
(No answer submitted)
Answer
\(g'(t) = \dfrac{-2(5t + 1) - (3 - 2t)(5)}{(5t + 1)^2} = \dfrac{13}{(5t + 1)^2}\)
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No explanation
36
Differentiation - Quotient Rule
Error Rate 100%
Wrong
Use the Quotient Rule to find the derivative of the function.
\(G(u) = \dfrac{6u^4 - 5u}{u + 1}\)
(No answer submitted)
Answer
\(G'(u) = \dfrac{(24u^3 - 5)(u + 1) - (6u^4 - 5u)}{(u + 1)^2}\)
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No explanation
37
Differentiation - Mixed
Error Rate 100%
Wrong
Differentiate.
\(f(t) = \dfrac{5t}{t^3 - t - 1}\)
(No answer submitted)
Answer
\(f'(t) = \dfrac{5(t^3 - t - 1) - 5t(3t^2 - 1)}{(t^3 - t - 1)^2}\)
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No explanation
38
Differentiation - Mixed
Error Rate 100%
Wrong
Differentiate.
\(F(x) = \dfrac{1}{2x^3 - 6x^2 + 5}\)
(No answer submitted)
Answer
\(F'(x) = \dfrac{-(6x^2 - 12x)}{(2x^3 - 6x^2 + 5)^2}\)
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No explanation
39
Differentiation - Mixed
Error Rate 100%
Wrong
Differentiate.
\(y = \dfrac{s - \sqrt{s}}{s^2}\)
(No answer submitted)
Answer
\(y' = -s^{-2} + \dfrac{3}{2} s^{\dfrac{-5}{2}}\)
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No explanation
40
Differentiation - Mixed
Error Rate 100%
Wrong
Differentiate.
\(y = \dfrac{\sqrt{x}}{\sqrt{x} + 1}\)
(No answer submitted)
Answer
\(y' = \dfrac{1}{2 \sqrt{x}(\sqrt{x} + 1)^2}\)
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No explanation
41
Differentiation - Mixed
Error Rate 100%
Wrong
Differentiate.
\(F(x) = \dfrac{2x^5 + x^4 - 6x}{x^3}\)
(No answer submitted)
Answer
\(F'(x) = 2 + x^{-2} + 12x^{-3}\)
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No explanation
42
Differentiation - Mixed
Error Rate 100%
Wrong
Differentiate.
\(y = \dfrac{(u + 2)^2}{1 - u}\)
(No answer submitted)
Answer
\(y' = \dfrac{2(u + 2)(1 - u) + (u + 2)^2}{(1 - u)^2}\)
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No explanation
43
Differentiation - Mixed
Error Rate 100%
Wrong
Differentiate.
\(H(u) = (u - \sqrt{u})(u + \sqrt{u})\)
(No answer submitted)
Answer
\(H'(u) = 2u - 1\)
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No explanation
44
Differentiation - Mixed
Error Rate 100%
Wrong
Differentiate.
\(A(v) = v^{\dfrac{2}{3}}(2v^2 + 1 - v^{-2})\)
(No answer submitted)
Answer
\(A'(v) = \dfrac{2}{3} v^{\dfrac{-1}{3}}(2v^2 + 1 - v^{-2}) + v^{\dfrac{2}{3}}(4v + 2v^{-3})\)
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No explanation
45
Differentiation - Mixed
Error Rate 100%
Wrong
Differentiate.
\(J(u) = \left(\dfrac{1}{u} + \dfrac{1}{u^2}\right)\left(u + \dfrac{1}{u}\right)\)
(No answer submitted)
Answer
\(J'(u) = (-u^{-2} - 2u^{-3})(u + u^{-1}) + (u^{-1} + u^{-2})(1 - u^{-2})\)
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No explanation
46
Differentiation - Mixed
Error Rate 100%
Wrong
Differentiate.
\(h(w) = (w^2 + 3w)(w^{-1} - w^{-4})\)
(No answer submitted)
Answer
\(h'(w) = (2w + 3)(w^{-1} - w^{-4}) + (w^2 + 3w)(-w^{-2} + 4w^{-5})\)
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No explanation
47
Differentiation - Mixed
Error Rate 100%
Wrong
Differentiate.
\(f(t) = \dfrac{\sqrt[3]{t}}{t - 3}\)
(No answer submitted)
Answer
\(f'(t) = \dfrac{\dfrac{1}{3} t^{\dfrac{-2}{3}}(t - 3) - t^{\dfrac{1}{3}}}{(t - 3)^2}\)
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No explanation
48
Differentiation - Mixed
Error Rate 100%
Wrong
Differentiate.
\(y = \dfrac{c x}{1 + c x}\)
(No answer submitted)
Answer
\(y' = \dfrac{c}{(1 + c x)^2}\)
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No explanation
49
Differentiation - Mixed
Error Rate 100%
Wrong
Differentiate.
\(G(y) = \dfrac{B}{A y^3 + B}\)
(No answer submitted)
Answer
\(G'(y) = \dfrac{-3A B y^2}{(A y^3 + B)^2}\)
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No explanation
50
Differentiation - Mixed
Error Rate 100%
Wrong
Differentiate.
\(F(t) = \dfrac{A t}{B t^2 + C t^3}\)
(No answer submitted)
Answer
\(F'(t) = \dfrac{A(B t^2 + C t^3) - A t(2B t + 3C t^2)}{(B t^2 + C t^3)^2}\)
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No explanation
51
Differentiation - Mixed
Error Rate 100%
Wrong
Differentiate.
\(f(x) = \dfrac{x}{x + \dfrac{c}{x}}\)
(No answer submitted)
Answer
\(f'(x) = \dfrac{x^2 + c - x \cdot \dfrac{x^2 - c}{x^2}}{\left(x + \dfrac{c}{x}\right)^2}\)
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No explanation
52
Differentiation - Mixed
Error Rate 100%
Wrong
Differentiate.
\(f(x) = \dfrac{a x + b}{c x + d}\)
(No answer submitted)
Answer
\(f'(x) = \dfrac{a d - b c}{(c x + d)^2}\)
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No explanation
53
Differentiation - General Polynomial
Error Rate 100%
Wrong
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 \(P'(x)\).
(No answer submitted)
Answer
\(P'(x) = n a_n x^{n-1} + (n-1) a_{n-1} x^{n-2} + \cdots + 2a_2 x + a_1\)
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No explanation
54
Differentiation - Graph Comparison
Error Rate 100%
Wrong
Find \(f'(x)\). Compare the graphs of \(f\) and \(f'\) and use them to explain why your answer is reasonable.
\(f(x) = x^4 - 2x^3 + x^2\)
(No answer submitted)
Answer
\(f'(x) = 4x^3 - 6x^2 + 2x\)
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No explanation
55
Differentiation - Graph Comparison
Error Rate 100%
Wrong
Find \(f'(x)\). Compare the graphs of \(f\) and \(f'\) and use them to explain why your answer is reasonable.
\(f(x) = 3x^{15} - 5x^3 + 3\)
(No answer submitted)
Answer
\(f'(x) = 45x^{14} - 15x^2\)
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No explanation
56
Differentiation - Graph Comparison
Error Rate 100%
Wrong
Find \(f'(x)\). Compare the graphs of \(f\) and \(f'\) and use them to explain why your answer is reasonable.
\(f(x) = x + \dfrac{1}{x}\)
(No answer submitted)
Answer
\(f'(x) = 1 - \dfrac{1}{x^2}\)
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No explanation
57
Differentiation - Graph Comparison
Error Rate 100%
Wrong
(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).
(No answer submitted)
Answer
\(f'(x) = 4x^3 - 9x^2 - 12x + 7\)
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No explanation
58
Differentiation - Graph Comparison
Error Rate 100%
Wrong
(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).
(No answer submitted)
Answer
\(g'(x) = \dfrac{2x}{(x^2 + 1)^2}\)
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No explanation
59
Differentiation - Tangent Line
Error Rate 100%
Wrong
Find an equation of the tangent line to the curve at the given point.
\(y = \dfrac{2x}{x + 1}\), \((1, 1)\)
(No answer submitted)
Answer
\(y - 1 = \dfrac{1}{2}(x - 1)\)
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No explanation
60
Differentiation - Tangent Line
Error Rate 100%
Wrong
Find an equation of the tangent line to the curve at the given point.
\(y = 2x^3 - x^2 + 2\), \((1, 3)\)
(No answer submitted)
Answer
\(y - 3 = 4(x - 1)\)
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No explanation
61
Differentiation - Tangent Line
Error Rate 100%
Wrong
Find equations of the tangent line and normal line to the curve at the given point.
\(y = x + \sqrt{x}\), \((1, 2)\)
(No answer submitted)
Answer
Tangent: \(y - 2 = \dfrac{3}{2}(x - 1)\); Normal: \(y - 2 = \dfrac{-2}{3}(x - 1)\)
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No explanation
62
Differentiation - Tangent Line
Error Rate 100%
Wrong
Find equations of the tangent line and normal line to the curve at the given point.
\(y = x^{\dfrac{3}{2}}\), \((1, 1)\)
(No answer submitted)
Answer
Tangent: \(y - 1 = \dfrac{3}{2}(x - 1)\); Normal: \(y - 1 = \dfrac{-2}{3}(x - 1)\)
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No explanation
63
Differentiation - Tangent Line
Error Rate 100%
Wrong
Find equations of the tangent line and normal line to the curve at the given point.
\(y = \dfrac{3x}{1 + 5x^2}\), \(\left(1, \dfrac{1}{2}\right)\)
(No answer submitted)
Answer
Tangent: \(y - \dfrac{1}{2} = \dfrac{-7}{36}(x - 1)\); Normal: \(y - \dfrac{1}{2} = \dfrac{36}{7}(x - 1)\)
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No explanation
64
Differentiation - Tangent Line
Error Rate 100%
Wrong
Find equations of the tangent line and normal line to the curve at the given point.
\(y = \dfrac{\sqrt{x}}{x + 1}\), \((4, 0.4)\)
(No answer submitted)
Answer
Tangent: \(y - 0.4 = \dfrac{-3}{100}(x - 4)\); Normal: \(y - 0.4 = \dfrac{100}{3}(x - 4)\)
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No explanation
65
Differentiation - Tangent Line
Error Rate 100%
Wrong
(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.
(No answer submitted)
Answer
\(y - \dfrac{1}{2} = \dfrac{1}{2}(x + 1)\)
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No explanation
66
Differentiation - Tangent Line
Error Rate 100%
Wrong
(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.
(No answer submitted)
Answer
\(y - 0.3 = \dfrac{-8}{100}(x - 3)\)
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No explanation
67
Differentiation - Higher Derivatives
Error Rate 100%
Wrong
Find the first and second derivatives of the function.
\(f(x) = 0.001x^5 - 0.02x^3\)
(No answer submitted)
Answer
\(f'(x) = 0.005x^4 - 0.06x^2\), \(f''(x) = 0.02x^3 - 0.12x\)
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No explanation
68
Differentiation - Higher Derivatives
Error Rate 100%
Wrong
Find the first and second derivatives of the function.
\(G(r) = \sqrt{r} + \sqrt[3]{r}\)
(No answer submitted)
Answer
\(G'(r) = \dfrac{1}{2} r^{\dfrac{-1}{2}} + \dfrac{1}{3} r^{\dfrac{-2}{3}}\), \(G''(r) = -\dfrac{1}{4} r^{\dfrac{-3}{2}} - \dfrac{2}{9} r^{\dfrac{-5}{3}}\)
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No explanation
69
Differentiation - Higher Derivatives
Error Rate 100%
Wrong
Find the first and second derivatives of the function.
\(f(x) = \dfrac{x^2}{1 + 2x}\)
(No answer submitted)
Answer
\(f'(x) = \dfrac{2x(1 + 2x) - 2x^2}{(1 + 2x)^2} = \dfrac{2x + 2x^2}{(1 + 2x)^2}\)
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No explanation
70
Differentiation - Higher Derivatives
Error Rate 100%
Wrong
Find the first and second derivatives of the function.
\(f(x) = \dfrac{1}{3 - x}\)
(No answer submitted)
Answer
\(f'(x) = \dfrac{1}{(3 - x)^2}\), \(f''(x) = \dfrac{2}{(3 - x)^3}\)
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No explanation
71
Differentiation - Higher Derivatives
Error Rate 100%
Wrong
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 \(f''\).
\(f(x) = 2x - 5x^{\dfrac{3}{4}}\)
(No answer submitted)
Answer
\(f'(x) = 2 - \dfrac{15}{4} x^{\dfrac{-1}{4}}\), \(f''(x) = \dfrac{15}{16} x^{\dfrac{-5}{4}}\)
Compare with the answer above and grade yourself:
No explanation
72
Differentiation - Higher Derivatives
Error Rate 100%
Wrong
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 \(f''\).
\(f(x) = \dfrac{x^2 - 1}{x^2 + 1}\)
(No answer submitted)
Answer
\(f'(x) = \dfrac{4x}{(x^2 + 1)^2}\)
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No explanation
73
Differentiation - Motion
Error Rate 100%
Wrong
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.
(No answer submitted)
Answer
(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
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No explanation
74
Differentiation - Motion
Error Rate 100%
Wrong
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.
(No answer submitted)
Answer
(a) \(v(t) = 4t^3 - 6t^2 + 2t - 1\), \(a(t) = 12t^2 - 12t + 2\); (b) \(a(1) = 2\) m/s^2
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No explanation
75
Differentiation - Applied
Error Rate 100%
Wrong
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.
(No answer submitted)
Answer
\(\dfrac{d L}{d A} = 0.0465 A^2 - 0.744 A + 3.95\); at \(A = 12\): \(\dfrac{d L}{d A} bar.v_{A=12} \approx 0.742\) inches per year
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No explanation
76
Differentiation - Applied
Error Rate 100%
Wrong
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.
(No answer submitted)
Answer
\(S'(A) = 0.882 \cdot 0.842 A^{-0.158} = 0.742644 A^{-0.158}\); \(S'(100) \approx 0.36\)
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No explanation
77
Differentiation - Applied
Error Rate 100%
Wrong
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?
(No answer submitted)
Answer
(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
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No explanation
78
Differentiation - Applied
Error Rate 100%
Wrong
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.
(No answer submitted)
Answer
Answers depend on the quadratic regression model fitted to the data.
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No explanation
79
Differentiation - Given Values
Error Rate 100%
Wrong
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) \(\left(\dfrac{g}{f}\right)'(5)\)
(No answer submitted)
Answer
(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) \(\left(\dfrac{g}{f}\right)'(5) = \dfrac{g'(5) f(5) - g(5) f'(5)}{[f(5)]^2} = 20\)
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No explanation
80
Differentiation - Given Values
Error Rate 100%
Wrong
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) \(h(x) = \dfrac{g(x)}{f(x) + g(x)}\)
(No answer submitted)
Answer
(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) \(h'(4) = \dfrac{-3(2+5) - 5(6+(-3))}{(2+5)^2} = \dfrac{-36}{49}\)
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No explanation
81
Differentiation - Given Values
Error Rate 100%
Wrong
If \(f(x) = \sqrt{x} \cdot g(x)\), where \(g(4) = 8\) and \(g'(4) = 7\), find \(f'(4)\).
(No answer submitted)
Answer
\(f'(4) = \dfrac{1}{2 \sqrt{4}} g(4) + \sqrt{4} g'(4) = \dfrac{8}{4} + 2(7) = 2 + 14 = 16\)
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No explanation
82
Differentiation - Given Values
Error Rate 100%
Wrong
If \(h(2) = 4\) and \(h'(2) = -3\), find \(\dfrac{d}{d x}(\dfrac{h(x)}{x}) bar.v_{x = 2}\).
(No answer submitted)
Answer
\(\dfrac{d}{d x}(\dfrac{h(x)}{x}) = \dfrac{h'(x) x - h(x)}{x^2}\); at \(x = 2\): \(\dfrac{-3(2) - 4}{4} = \dfrac{-10}{4} = \dfrac{-5}{2}\)
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No explanation
83
Differentiation - Given Values
Error Rate 100%
Wrong
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 \(v'(4)\).
(No answer submitted)
Answer
Read values from graph to apply product and quotient rules.
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No explanation
84
Differentiation - Given Values
Error Rate 100%
Wrong
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 \(Q'(7)\).
(No answer submitted)
Answer
Read values from graph to apply product and quotient rules.
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No explanation
85
Differentiation - Abstract/General
Error Rate 100%
Wrong
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) \(y = \dfrac{g(x)}{x}\)
(No answer submitted)
Answer
(a) \(y' = g(x) + x g'(x)\); (b) \(y' = \dfrac{g(x) - x g'(x)}{[g(x)]^2}\); (c) \(y' = \dfrac{x g'(x) - g(x)}{x^2}\)
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No explanation
86
Differentiation - Abstract/General
Error Rate 100%
Wrong
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) \(y = \dfrac{1 + x f(x)}{\sqrt{x}}\)
(No answer submitted)
Answer
(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) \(y' = \dfrac{f(x) + x f'(x)}{\sqrt{x}} - \dfrac{1 + x f(x)}{2x^{\dfrac{3}{2}}}\)
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No explanation
87
Differentiation - Finding Points
Error Rate 100%
Wrong
Find the points on the curve \(y = x^3 + 3x^2 - 9x + 10\) where the tangent is horizontal.
(No answer submitted)
Answer
\(y' = 3x^2 + 6x - 9 = 3(x + 3)(x - 1) = 0\) gives \(x = -3\) and \(x = 1\). Points: \((-3, 37)\) and \((1, 5)\).
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No explanation
88
Differentiation - Finding Points
Error Rate 100%
Wrong
For what values of \(x\) does the graph of \(f(x) = x^3 + 3x^2 + x + 3\) have a horizontal tangent?
(No answer submitted)
Answer
\(f'(x) = 3x^2 + 6x + 1 = 0\) gives \(x = \dfrac{-6 \pm \sqrt{24}}{6} = \dfrac{-3 \pm \sqrt{6}}{3}\)
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No explanation
89
Differentiation - Finding Points
Error Rate 100%
Wrong
Show that the curve \(y = 6x^3 + 5x - 3\) has no tangent line with slope 4.
(No answer submitted)
Answer
\(y' = 18x^2 + 5 \geq 5 > 4\) for all \(x\). Since \(y' > 4\) always, no tangent line has slope 4.
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No explanation
90
Differentiation - Finding Points
Error Rate 100%
Wrong
Find an equation of the tangent line to the curve \(y = x^4 + 1\) that is parallel to the line \(32x - y = 15\).
(No answer submitted)
Answer
Slope = 32. \(y' = 4x^3 = 32\) gives \(x = 2\). Point: \((2, 17)\). Tangent: \(y - 17 = 32(x - 2)\).
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No explanation
91
Differentiation - Finding Points
Error Rate 100%
Wrong
Find equations of both lines that are tangent to the curve \(y = x^3 - 3x^2 + 3x - 3\) and are parallel to the line \(3x - y = 15\).
(No answer submitted)
Answer
Slope = 3. \(y' = 3x^2 - 6x + 3 = 3(x - 1)^2 = 3\) gives \(x = 0\) or \(x = 2\). Two tangent lines.
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No explanation
92
Differentiation - Finding Points
Error Rate 100%
Wrong
Find equations of the tangent lines to the curve \(y = \dfrac{x - 1}{x + 1}\) that are parallel to the line \(x - 2y = 2\).
(No answer submitted)
Answer
Slope = \(\dfrac{1}{2}\). \(y' = \dfrac{2}{(x + 1)^2} = \dfrac{1}{2}\) gives \((x + 1)^2 = 4\), so \(x = 1\) or \(x = -3\).
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No explanation
93
Differentiation - Finding Points
Error Rate 100%
Wrong
Find an equation of the normal line to the parabola \(y = \sqrt{x}\) that is parallel to the line \(2x + y = 1\).
(No answer submitted)
Answer
Normal slope = \(-2\). Tangent slope = \(\dfrac{1}{2}\). \(y' = \dfrac{1}{2 \sqrt{x}} = \dfrac{1}{2}\) gives \(x = 1\), \(y = 1\). Normal: \(y - 1 = -2(x - 1)\).
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No explanation
94
Differentiation - Finding Points
Error Rate 100%
Wrong
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.
(No answer submitted)
Answer
Tangent slope at \((-1, 0)\): \(y' = 2x\), so slope = \(-2\). Normal slope = \(\dfrac{1}{2}\). Normal: \(y = \dfrac{1}{2}(x + 1)\). Solve with \(y = x^2 - 1\).
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No explanation
95
Differentiation - Finding Points
Error Rate 100%
Wrong
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.
(No answer submitted)
Answer
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 \((-2, 4)\).
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No explanation
96
Differentiation - Finding Points
Error Rate 100%
Wrong
(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.
(No answer submitted)
Answer
(a) Tangent at \((a, a^2 + a)\) with slope \(2a + 1\) passes through \((2, -3)\). Solve to find \(a\).
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No explanation
97
Differentiation - Finding Points
Error Rate 100%
Wrong
For what values of \(a\) and \(b\) is the line \(2x + y = b\) tangent to the parabola \(y = a x^2\) when \(x = 2\)?
(No answer submitted)
Answer
At \(x = 2\): slope of tangent = \(2a(2) = 4a = -2\), so \(a = \dfrac{-1}{2}\). Then \(y = \dfrac{-1}{2}(4) = -2\), and \(b = 2(2) + (-2) = 2\).
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No explanation
98
Differentiation - Patterns/nth
Error Rate 100%
Wrong
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) \(f(x) = \dfrac{1}{x}\)
(No answer submitted)
Answer
(a) \(f^{(n)}(x) = n!\); (b) \(f^{(n)}(x) = \dfrac{(-1)^n n!}{x^{n+1}}\)
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No explanation
99
Differentiation - Patterns/nth
Error Rate 100%
Wrong
Find a second-degree polynomial \(P\) such that \(P(2) = 5\), \(P'(2) = 3\), and \(P''(2) = 2\).
(No answer submitted)
Answer
Let \(P(x) = a x^2 + b x + c\). \(P''(x) = 2a = 2\), so \(a = 1\). \(P'(x) = 2x + b\), \(P'(2) = 4 + b = 3\), \(b = -1\). \(P(2) = 4 - 2 + c = 5\), \(c = 3\). \(P(x) = x^2 - x + 3\).
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100
Differentiation - Proof/Theory
Error Rate 100%
Wrong
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.)
(No answer submitted)
Answer
\(y' = 2A x + B\), \(y'' = 2A\). 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 \(A = \dfrac{-1}{2}\), \(B = \dfrac{-1}{2}\), \(C = \dfrac{-3}{4}\).
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No explanation
101
Differentiation - Proof/Theory
Error Rate 100%
Wrong
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 \((2, 0)\).
(No answer submitted)
Answer
\(y' = 3a x^2 + 2b x + c\). Conditions: \(y'(-2) = 0\), \(y'(2) = 0\), \(y(-2) = 6\), \(y(2) = 0\). Solve the system.
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No explanation
102
Differentiation - Proof/Theory
Error Rate 100%
Wrong
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 \((2, 15)\).
(No answer submitted)
Answer
\(y' = 2a x + b\). \(y'(1) = 2a + b = 4\), \(y'(-1) = -2a + b = -8\). Solving: \(a = 3\), \(b = -2\). \(y(2) = 12 - 4 + c = 15\), \(c = 7\). \(y = 3x^2 - 2x + 7\).
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No explanation
103
Differentiation - Applied
Error Rate 100%
Wrong
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.
(No answer submitted)
Answer
Total income \(= P \cdot I\). Rate \(= P' \cdot I + P \cdot I' = 1300 \cdot 62370 + 108250 \cdot 2500 \approx \$351,706,000\) per year.
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No explanation
104
Differentiation - Applied
Error Rate 100%
Wrong
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.
(No answer submitted)
Answer
(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.
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No explanation
105
Differentiation - Applied
Error Rate 100%
Wrong
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.
(No answer submitted)
Answer
\(\dfrac{d v}{d [S]} = \dfrac{0.14 \cdot 0.015}{(0.015 + [S])^2} = \dfrac{0.0021}{(0.015 + [S])^2}\)
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No explanation
106
Differentiation - Applied
Error Rate 100%
Wrong
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.
(No answer submitted)
Answer
\(B'(t) = N'(t) w(t) + N(t) w'(t)\). First term: biomass increase from population growth. Second term: biomass increase from individual weight gain.
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No explanation
107
Differentiation - Proof/Theory
Error Rate 100%
Wrong
(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 \(y = (x^4 + 3x^3 + 17x + 82)^3\).
(No answer submitted)
Answer
(a) Apply Product Rule to \(F = (f g) h\). (b) Set \(f = g = h\). (c) \(y' = 3(x^4 + 3x^3 + 17x + 82)^2(4x^3 + 9x^2 + 17)\).
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No explanation
108
Differentiation - Proof/Theory
Error Rate 100%
Wrong
(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.
(No answer submitted)
Answer
(a) Apply Quotient Rule with \(f(x) = 1\). (b) \(F'(x) = \dfrac{-(6x^2 - 12x)}{(2x^3 - 6x^2 + 5)^2}\).
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No explanation
109
Differentiation - Proof/Theory
Error Rate 100%
Wrong
Use the Product Rule to prove the Quotient Rule. [Hint: Write \(f(x) = [\dfrac{f(x)}{g(x)}] \cdot g(x)\).]
(No answer submitted)
Answer
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 \(F'(x)\).
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No explanation
110
Differentiation - Proof/Theory
Error Rate 100%
Wrong
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 \(F^{(n)}\).
(No answer submitted)
Answer
(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).
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No explanation