Calculus

Problem 1101

Find the limit: $\lim _{x \rightarrow-\infty} \frac{x^{6}-3 x^{4}}{2 x^{2}-2 x+1}$ .

See Solution

Problem 1102

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{3x}{x+3}$ , with $h \neq 0$ . Simplify your answer.

See Solution

Problem 1103

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{10}{x^{2}}$ , where $h \neq 0$ . Simplify your answer.

See Solution

Problem 1104

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{11}{x^{2}}$ , simplifying your answer.

See Solution

Problem 1105

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{5 x}{x+6}$ , where $h \neq 0$ . Simplify your answer.

See Solution

Problem 1106

A rifle fires bullets with speed $v=(-5.35 \times 10^{7}) t^{2}+(2.30 \times 10^{5}) t$ . Find acceleration, position, time, speed, and barrel length.

See Solution

Problem 1107

Which table shows values of $g$ with $\lim_{x \to 7} g(x) = 6$ ? Options: (A), (B), (C), (D).

See Solution

Problem 1108

What does $\lim _{x \rightarrow 3} f(x)=5$ mean? Choose the correct interpretation: (A), (B), (C), or (D).

See Solution

Problem 1109

A car's distance is $s(t)$ in feet after $t$ seconds. Estimate its velocity at $t=6$ using options (A) to (D).

See Solution

Problem 1110

Find $\lim _{x \rightarrow 2^{+}} f(x)$ for the piecewise function: $f(x)=\{5x-3 \text{ if } x<2, 9 \text{ if } x=2, 4x+3 \text{ if } x>2\}$ .

See Solution

Problem 1111

Given the table of values for $f(x)$ at $x$ near 4, what limit conclusion is supported? (A) $\lim _{x \rightarrow 4} f(x)=6$ , (B) $\lim _{x \rightarrow 4} f(x)=7$ , (C) $\lim _{x \rightarrow 4^{-}} f(x)=6$ and $\lim _{x \rightarrow 4^{+}} f(x)=7$ , (D) $\lim _{x \rightarrow 4^{-}} f(x)=7$ and $\lim _{x \rightarrow 4^{+}} f(x)=6$ .

See Solution

Problem 1112

Given the table of $x$ and $f(x)$ values, which limit conclusion about $f(x)$ as $x$ approaches 6 is correct? (A) $\lim _{x \rightarrow 6} f(x)=0$ (B) $\lim _{x \rightarrow 6} f(x)=6$ (C) $\lim _{x \rightarrow 6} f(x)=10$ (D) $\lim _{x \rightarrow 6} f(x)$ does not exist.

See Solution

Problem 1113

Given the table values of $f(x)$ for $x$ near 4, which limit conclusion is correct? (A) $\lim _{x \rightarrow 4} f(x)=6$ (B) $\lim _{x \rightarrow 4} f(x)=7$ (C) $\lim _{x \rightarrow 4^{-}} f(x)=6$ and $\lim _{x \rightarrow 4^{+}} f(x)=7$ (D) $\lim _{x \rightarrow 4^{-}} f(x)=7$ and $\lim _{x \rightarrow 4^{+}} f(x)=6$

See Solution

Problem 1114

Find $\lim _{x \rightarrow 2}(h(x)(5 f(x)+g(x)))$ given $f(2)=3$ , $g(2)=-6$ , $h(2)=-3$ , limits at $x=2$ .

See Solution

Problem 1115

Find $\lim _{x \rightarrow 5}(f(-x)+3 g(x))$ given $\lim _{x \rightarrow-5} f(x)=4$ , $\lim _{x \rightarrow 5} f(x)=2$ , $\lim _{x \rightarrow 5} g(x)=5$ .

See Solution

Problem 1116

Find $\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}-2 x-15}$ . Options: (A) 0 (B) $\frac{3}{5}$ (C) $\frac{3}{4}$ (D) 1 (E) nonexistent.

See Solution

Problem 1117

Find $\lim _{x \rightarrow 0} \frac{\tan (2 x)}{6 x \sec (3 x)}$ . Choose from (A) 0, (B) $\frac{1}{6}$ , (C) $\frac{1}{3}$ , (D) nonexistent.

See Solution

Problem 1118

Find $\lim _{x \rightarrow 2} f(x)$ for the function $f(x)=\frac{x^{2}-4}{x^{2}+x-6}$ . Choices: (A) 0, (B) $\frac{2}{3}$ , (C) $\frac{4}{5}$ , (D) nonexistent.

See Solution

Problem 1119

Find the value of $k$ such that $\lim _{x \rightarrow 2} f(x)=3$ for $f(x)=\frac{(x-2)(x^{2}-k^{2})}{(x^{2}-4)(x-k)}$ .

See Solution

Problem 1120

Find $\lim _{x \rightarrow 1} f(x)$ for $f(x)=\frac{x^{2}-1}{\sqrt{x}-1}$ . Choices: (A) 4, (B) 2, (C) 0, (D) nonexistent.

See Solution

Problem 1121

Find $\lim _{x \rightarrow \frac{\pi}{4}} g(x)$ for $g(x)=\frac{\cos x-\sin x}{1-2 \sin ^{2} x}$ . Options: (A) 0 (B) $\frac{1}{\sqrt{2}}$ (C) $\sqrt{2}$ (D) Limit does not exist.

See Solution

Problem 1122

Find the derivative of $y=\frac{1}{x}$ , calculate $\lim_{x \to \infty} \frac{-4x}{7x-3}$ , and find two numbers with a difference of 4 that minimize their product.

See Solution

Problem 1123

Find the average rate of change of $f(x)=\sin x$ from $x_{1}=\frac{\pi}{4}$ to $x_{2}=\frac{3\pi}{2}$ .

See Solution

Problem 1124

Differentiate the following using the Product Rule:
1. $y=x^{2}(x+1)$
2. $y=(x^{2}+3)(x+6)$
3. $y=\sqrt{x}(x^{3}+6)$
4. $y=(2 x^{2}+4 x-3)(3 x+4)$

See Solution

Problem 1125

Differentiate these functions using the quotient rule: $y=\frac{x+2}{x+3}$ , $y=\frac{1}{x(2x+1)}$ , $y=\frac{4x+1}{3x+8}$ , $y=\frac{2-3x}{1+x}$ .

See Solution

Problem 1126

Find the limit as $x$ approaches 2 for the expression $x^{2} - 4x + 3$ .

See Solution

Problem 1127

Find the limit as $x$ approaches 3 for the expression $\frac{4x+2}{x+4}$ .

See Solution

Problem 1128

Find the limit: $\lim _{x \rightarrow \frac{\pi}{3}} \frac{\sin 2 x}{\sin x}$ .

See Solution

Problem 1129

Calculate the limit: $\lim _{x \rightarrow 0}\left(\frac{\sqrt{1-\cos 2 x}}{\sqrt{2} x}\right)$ .

See Solution

Problem 1130

Determina la velocidad de una moneda en caída libre tras $2.0 \mathrm{~s}$ desde reposo en la Torre Colpatria.

See Solution

Problem 1131

Calculate the difference quotient $\frac{f(x+h)-f(x)}{h}$ for the function $f(x)=-7x+6$ , where $h \neq 0$ .

See Solution

Problem 1132

Find the derivatives of the following functions:
1. $f(x)=x^{4}-5 x^{3}+9 x^{2}-7 x+5$
2. $y=\frac{3 x^{2}-6 x}{x^{3}}$
3. $g(x)=8 \sqrt{x}-\frac{6}{\sqrt[3]{x^{2}}}$
4. $h(x)=\frac{3 x+2}{x^{2}+1}$
5. $f(x)=3 x(8 x-3)^{7}$
6. $f(x)=3 x^{4}-4 x^{2}+7 x-11$
7. $k(x)=x \sqrt{9-x^{2}}$
8. $f(x)=-3\left(2 x^{2}-5 x+1\right)$
9. $g(x)=\frac{x^{2}}{3}-\frac{3}{x^{2}}$
10. $y=\frac{9}{x^{2}}-\frac{8}{x^{3}}+\frac{2}{x^{4}}$
11. $y=\sqrt[3]{x^{2}-3 x+4}$
12. $y=5 x^{2}-5 \sqrt{x}-\frac{3}{x}$
13. $y=\sqrt{x}-\frac{1}{\sqrt{x}}$
14. $y=(3 x-2)(2 x+1)$
15. $y=\frac{\sqrt{x}+1}{\sqrt{x}-1}$
16. $f(x)=4 x^{3}-3 x^{2}+2 x$
17. $y=z^{5}-e^{z} \ln z$
18. $f(x)=2 e^{x}-8^{x}$

See Solution

Problem 1133

Find the derivatives of these functions: 1. $f(x)=x^{4}-5 x^{3}+9 x^{2}-7 x+5$ ; 2. $g(x)=8 \sqrt{x}-\frac{6}{\sqrt[3]{x^{2}}}$ ; 3. $h(x)=\frac{3 x+2}{x^{2}+1}$ ; 4. $f(x)=3 x(8 x-3)^{7}$ ; 5. $f(x)=3 x^{4}-4 x^{2}+7 x-11$ ; 6. $k(x)=x \sqrt{9-x^{2}}$ ; 7. $f(x)=-3(2 x^{2}-5 x+1)$ ; 8. $g(x)=\frac{x^{2}}{3}-\frac{3}{x^{2}}$ ; 9. $y=\frac{9}{x^{2}}-\frac{8}{x^{3}}+\frac{2}{x^{4}}$ .

See Solution

Problem 1134

A golf ball is hit at 130 ft/s at $45^{\circ}$ . Find the distance to max height using $h(x)=\frac{-32 x^{2}}{130^{2}}+x$ .

See Solution

Problem 1135

The expression $\frac{f(x+h)-f(x)}{h}$ is called the derivative of $f$ .

See Solution

Problem 1136

Find the derivatives of the following functions:
10. $y=\frac{3 x^{2}-6 x}{x^{3}}$
11. $y=\sqrt[3]{x^{2}-3 x+4}$
12. $y=5 x^{2}-5 \sqrt{x}-\frac{3}{x}$
13. $y=\sqrt{x}-\frac{1}{\sqrt{x}}$
14. $y=(3 x-2)(2 x+1)$
15. $y=\frac{\sqrt{x}+1}{\sqrt{x}-1}$
16. $f(x)=4 x^{3}-3 x^{2}+2 x$
17. $y=z^{5}-e^{z} \ln z$
18. $f(x)=2 e^{x}-8^{x}$

See Solution

Problem 1137

Find the rate of change of $y=\cos 5 x$ . Choices: A. $\sin 5 x$ , B. $5 \sin 5 x$ , C. $-\sin 5 x$ , D. $-5 \sin 5 x$ .

See Solution

Problem 1138

Find the derivative of $f(x) = 5x^2$ .

See Solution

Problem 1139

Find the integral of $\frac{1-x^{2}}{x(1-2 x)}$ and simplify to get the final expression.

See Solution

Problem 1140

Calculate the integral $\int_{0}^{3}(1-e^{-x}) \, dx$ with $h=2$ .

See Solution

Problem 1141

Calculate the integral $\int_{v}^{t} t \cdot \cos \left(\frac{t}{5}\right) \cdot d t$ .

See Solution

Problem 1142

Show that $f(x)=x^{2}-6x+1$ on [1,3] meets Lagrange's theorem. Find where the tangent is parallel to the line joining A(1,-4) and B(3,-8).

See Solution

Problem 1143

Find the derivative of $y=x^{2}+3(4x^{2}+2)^{4}$ .

See Solution

Problem 1144

Find the derivatives of these functions: 1. $y=e^{-x}$ , 2. $y=e^{x+1}$ , 3. $y=\frac{1}{e^{5x}}$ .

See Solution

Problem 1145

Find the tangent line equation for the function $f(x)=e^{-x} \ln (x)$ at the point $(1,0)$ .

See Solution

Problem 1146

Find the average rate of change of $D(t)$ from $t=0.2$ to $t=0.4$ . Use units of miles per hour.

See Solution

Problem 1147

A cube's volume increases at $3 \frac{1}{2} \mathrm{~cm}^{3} \mathrm{~s}^{-1}$ . Find the base's change rate when length is $6 \mathrm{~cm}$ .

See Solution

Problem 1148

Find the derivative, integral, domain, range, and solution for the function $g(x)=\ln(2x-4)$ .

See Solution

Problem 1149

Evaluate the limit using the Squeeze Theorem: $\lim _{x \rightarrow 0} x^{2} \cos \left(\frac{1}{x^{2}}\right)$ .

See Solution

Problem 1150

Weekly cost to make $x$ cars and $y$ trucks is $C(x, y)=240,000+6,000x+4,000y$ . Find marginal costs and describe graphs.

See Solution

Problem 1151

Which statements are true for all $c$ ? I. $\lim _{x \rightarrow c} f(x)=0 \Rightarrow \lim _{x \rightarrow c}|f(x)|=0$ . II. $\lim _{x \rightarrow c}|f(x)|=0 \Rightarrow \lim _{x \rightarrow c} f(x)=0$ .

See Solution

Problem 1152

Determine the truth of these statements for all $c$ and $L$ : I. $\lim _{x \rightarrow c} f(x)=L \Rightarrow \lim _{x \rightarrow c}|f(x)|=|L|$ ; II. $\lim _{x \rightarrow c}|f(x)|=|L| \Rightarrow \lim _{x \rightarrow c} f(x)=L$ .

See Solution

Problem 1153

Find all values of $a$ where $\lim_{x \to a} f(x)$ exists for the piecewise function $f(x)$ defined above.

See Solution

Problem 1154

Determine if the limits $L_{1} = \lim _{x \rightarrow 0+} \sin(\frac{\pi}{x})$ and $L_{2} = \lim _{x \rightarrow 0-} \sin(\frac{\pi}{x})$ exist.

See Solution

Problem 1155

Find the average velocity of a ball thrown upward on the moon, given $y(t)=25t-3t^2$ , over the interval from 1 to $1+h$ .

See Solution

Problem 1156

A ball is thrown with a velocity of $40 \mathrm{ft/s}$ . Find average velocity from $t=2$ for (a) 0.5s, 0.05s, 0.1s, 0.01s and (b) instantaneous velocity at $t=2$ .

See Solution

Problem 1157

Find the limit as $x$ approaches -1 for the expression $\frac{x^{2}+9 x+8}{8 x+8}$ .

See Solution

Problem 1158

Find the limit as $x$ approaches -8 for the expression $\frac{x^{2}-64}{4 x+32}$ .

See Solution

Problem 1159

Find the limit as $x$ approaches 5 for the expression $\frac{x^{2}-13 x+40}{x^{2}-8 x+15}$ .

See Solution

Problem 1160

Find $\lim _{x \rightarrow 0} \frac{3 x^{2}-\sin 2 x}{2 x-\sin 3 x}$ . What is the result? A) -1 B) 0 C) 2 D) Does not exist.

See Solution

Problem 1161

Find the limit: $\lim _{x \rightarrow 3^{-}} \frac{2-x}{x-3}$ . What is the result? A) $-\infty$ B) $+\infty$ C) 0 D) 1

See Solution

Problem 1162

Find the limit: $\lim _{x \rightarrow 0} \frac{4 \sin x}{2-2 \cos x}$ . What is the result? A) 2 B) $\pi$ C) 0 D) does not exist.

See Solution

Problem 1163

Find $\lim _{x \rightarrow c}[2 f(x)+3 g(x)]$ given $\lim _{x \rightarrow c} f(x)=\frac{-2}{3}$ and $\lim _{x \rightarrow c} g(x)=\frac{5}{4}$ .

See Solution

Problem 1164

Determine which statement about the function $f(x)=\left\{\begin{array}{cr}|x-3| & x<0 \\ 6-3 x & 0 \leq x<3 \\ -(x-3)^{2} & x \geq 3\end{array}\right.$ is false: A) $\lim _{x \rightarrow 2} f(x)=0$ B) $\lim _{x \rightarrow 3^{-}} f(x)=-3$ C) $\lim _{x \rightarrow 0^{-}} f(x)=6$ D) $\lim _{x \rightarrow 3^{+}} f(x)=0$

See Solution

Problem 1165

Find the limit: $\lim _{x \rightarrow -5} \frac{x^{2}-25}{x^{2}-2x-35}$ .

See Solution

Problem 1166

Find the half-life of a substance decaying as $y = y_{0} e^{-0.019 t}$ . Round your answer to the nearest tenth.

See Solution

Problem 1167

Find the half-life of a substance decaying at $5.7\%$ per day. Round your answer to the nearest hundredth.

See Solution

Problem 1168

How long does it take for a bacteria population to double at a continuous growth rate of $1\%$ per hour? Round to the nearest hundredth.

See Solution

Problem 1169

Find the annual interest rate for an investment of \$3000 that grows to \$3240 in 2 years, compounded continuously.

See Solution

Problem 1170

A rifle fires $22 L R$ bullets with speed $v=(-5.35 \times 10^{7}) t^{2}+(2.30 \times 10^{5}) t$ .
(a) Find acceleration $a=\frac{d v}{d t}$ and position $x=\int v d t$ .
(b) Time of acceleration (s).
(c) Speed when leaving barrel (m/s).
(d) Length of barrel (m).

See Solution

Problem 1171

Is the function $f(x) = \frac{x^{2}-x-20}{x^{2}-9 x+20}$ for $x \neq 5$ and $f(5) = 9$ continuous at $x=5$ ?

See Solution

Problem 1172

Calculate the integral: $\int \frac{2 x}{x^{2}-1} d x$

See Solution

Problem 1173

Find the difference quotient of $f(x)=x^{2}+4$ : calculate $\frac{f(x+h)-f(x)}{h}$ , $h \neq 0$ , and simplify.

See Solution

Problem 1174

Find the output level $x$ that minimizes the marginal cost $C(x)=x^{2}-140x+7100$ and determine the minimum cost.

See Solution

Problem 1175

Given a continuous function $f$ on $[-2,2]$ with $f(-2)=1$ and $f(2)=-1$ , which properties hold by the Intermediate Value Theorem? A. $f(c)=0$ for some $c$ in $(-1,1)$ ; B. $f(x)+1 \geq 0$ on $(-2,2)$ ; C. $f^{2}(c) \leq 1$ for all $c$ in $(-2,2)$ .

See Solution

Problem 1176

Find the limit $\lim _{x \rightarrow 7^{+}} F(x)$ for the function $F(x)=\frac{x^{2}-49}{|x-7|}$ . Does it exist?

See Solution

Problem 1177

Find $\lim _{x \rightarrow-1^{+}} f(x)$ for the function $f(x)$ with a hole at $x=-5$ and a value at $x=-1$ .

See Solution

Problem 1178

Find the limit: $\lim _{x \rightarrow 2} \frac{2 f(x) g(x)}{3 f(x)-g(x)}$ given that $\lim _{x \rightarrow 2} f(x)=4, \quad \lim _{x \rightarrow 2} g(x)=-1.$

See Solution

Problem 1179

Find the average rate of change of the function $f(x)=2x^{2}+4$ over the interval $[-5,1]$ . Include units if needed.

See Solution

Problem 1180

Find the average rate of change of $f(x)=3 x^{2}-\frac{x}{2}$ over the interval $[5,10]$ . Specify units if needed.

See Solution

Problem 1181

Find the integral of the function: $\int 4 x \cos (2-3 x) d x$

See Solution

Problem 1182

Is the function $f(x)$ , defined as $f(x)=\ln \frac{\sqrt{x+4}-1}{3+x}$ for $x \neq -3$ and $f(-3)=2$ , continuous at $x=-3$ ?

See Solution

Problem 1183

Is the function $f(x) = \sqrt{x^{2}-4}$ continuous at $x=0$ ?

See Solution

Problem 1184

Check if the function $f(x)=\begin{cases} e^{x /(x+1)} & \text{if } x \neq-1 \\ e^{-1} & \text{if } x=-1 \end{cases}$ is continuous at $x=-1$ .

See Solution

Problem 1185

Is the function $f(x)=\frac{1}{x+1}$ continuous at $x=c$ for $c=1$ ?

See Solution

Problem 1186

Is the function $f(x)=\left\{\begin{array}{ll}\frac{\sin x}{x} & \text { if } x \neq 0 \\ 1 & \text { if } x=0\end{array}\right.$ continuous at $x=0$ ?

See Solution

Problem 1187

Is the function $f(x)=\tan \left(\frac{2}{\pi} x\right)$ continuous at $x=c$ , where $c=\frac{\pi}{2}$ ?

See Solution

Problem 1188

Is the function $f(x)=\left\{\begin{array}{ll}\frac{1}{\ln \sqrt{x}-\pi / 2} & \text { if } x>e^{\pi} \\ 0 & \text { if } x \leqslant e^{\pi}\end{array}\right.$ continuous at $x=e^{\pi}$ ?

See Solution

Problem 1189

Check if the function $f(x)=\left\{\begin{array}{ll}\frac{\sqrt{x^{2}+1}-1}{x^{2}} & \text { if } x \neq 0 \\ \frac{1}{2} & \text { if } x=0\end{array}\right.$ is continuous at $x=0$ .

See Solution

Problem 1190

Check if the function $f(x)$ is continuous at $x=0$ , where $f(x)=\begin{cases}\arctan \frac{1}{x} & \text{if } x>0 \\ x+\frac{\pi}{2} & \text{if } x \leq 0\end{cases}$ .

See Solution

Problem 1191

Is the function $f(x)=e^{\ln x}$ continuous at $x=0$ ?

See Solution

Problem 1192

Is the function $f(x)=\left\{\begin{array}{ll}\sqrt{\sin ^{2} x+\cos ^{2} x} & \text { if } x>1 \\ 1 & \text { if } x \leqslant 1\end{array}\right.$ continuous at $x=1$ ?

See Solution

Problem 1193

Is the function $f(x)=\left\{\begin{array}{ll}\frac{\sqrt{x+4}-2}{x} & \text { if } x>0 \\ x-0.25 & \text { if } x \leqslant 0\end{array}\right.$ continuous at $x = c$ ? Explain.

See Solution

Problem 1194

Check if the function $f(x)=\begin{cases} \frac{\sqrt{x+4}-2}{x} & \text{if } x > 0 \\ x-0.25 & \text{if } x \leq 0 \end{cases}$ is continuous at $x=0$ .

See Solution

Problem 1195

How long to double a bacteria population growing at $5.4\%$ per hour using continuous exponential growth? Round to the nearest hundredth.

See Solution

Problem 1196

Find the half-life of a radioactive substance with a decay rate of $4.3\%$ per day using continuous exponential decay.

See Solution

Problem 1197

Find the time for a bacteria population to double with a growth rate of $1.5\%$ per hour using continuous growth.

See Solution

Problem 1198

Compare the rates of change of $f(x)=\ln (x-3)$ at $x=4$ and $x=10$ . Use the graph of $f$ for justification.

See Solution

Problem 1199

Calculate the limit as $t$ approaches 4 for the expression $\frac{t^{2}-2t-8}{t-4}$ .

See Solution

Problem 1200

Sketch the graph of $f(x)$ , then find $\lim_{x \rightarrow 0} f(x)$ and $\lim_{x \rightarrow 2} f(x)$ . Function: $f(x)=\begin{cases} \sin x & \text{if } x<0 \\ x^{2} & \text{if } 0 \leq x<9 \\ x & \text{if } x \geq 2 \end{cases}$

See Solution

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337

Calculus

Problem 1101

Find the limit: lim⁡x→−∞x6−3x42x2−2x+1\lim _{x \rightarrow-\infty} \frac{x^{6}-3 x^{4}}{2 x^{2}-2 x+1}limx→−∞​2x2−2x+1x6−3x4​.

Problem 1102

Find the difference quotient f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ for f(x)=3xx+3f(x)=\frac{3x}{x+3}f(x)=x+33x​, with h≠0h \neq 0h=0. Simplify your answer.

Problem 1103

Find the difference quotient f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ for f(x)=10x2f(x)=\frac{10}{x^{2}}f(x)=x210​, where h≠0h \neq 0h=0. Simplify your answer.

Problem 1104

Find the difference quotient f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ for f(x)=11x2f(x)=\frac{11}{x^{2}}f(x)=x211​, simplifying your answer.

Problem 1105

Find the difference quotient f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ for f(x)=5xx+6f(x)=\frac{5 x}{x+6}f(x)=x+65x​, where h≠0h \neq 0h=0. Simplify your answer.

Problem 1106

A rifle fires bullets with speed v=(−5.35×107)t2+(2.30×105)tv=(-5.35 \times 10^{7}) t^{2}+(2.30 \times 10^{5}) tv=(−5.35×107)t2+(2.30×105)t. Find acceleration, position, time, speed, and barrel length.

Problem 1107

Which table shows values of ggg with lim⁡x→7g(x)=6\lim_{x \to 7} g(x) = 6limx→7​g(x)=6? Options: (A), (B), (C), (D).

Problem 1108

What does lim⁡x→3f(x)=5\lim _{x \rightarrow 3} f(x)=5limx→3​f(x)=5 mean? Choose the correct interpretation: (A), (B), (C), or (D).

Problem 1109

A car's distance is s(t)s(t)s(t) in feet after ttt seconds. Estimate its velocity at t=6t=6t=6 using options (A) to (D).

Problem 1110

Find lim⁡x→2+f(x)\lim _{x \rightarrow 2^{+}} f(x)limx→2+​f(x) for the piecewise function: f(x)={5x−3 if x<2,9 if x=2,4x+3 if x>2}f(x)=\{5x-3 \text{ if } x<2, 9 \text{ if } x=2, 4x+3 \text{ if } x>2\}f(x)={5x−3 if x<2,9 if x=2,4x+3 if x>2}.

Problem 1111

Problem 1112

Problem 1113

Problem 1114

Find lim⁡x→2(h(x)(5f(x)+g(x)))\lim _{x \rightarrow 2}(h(x)(5 f(x)+g(x)))limx→2​(h(x)(5f(x)+g(x))) given f(2)=3f(2)=3f(2)=3, g(2)=−6g(2)=-6g(2)=−6, h(2)=−3h(2)=-3h(2)=−3, limits at x=2x=2x=2.

Problem 1115

Problem 1116

Find lim⁡x→−3x2−9x2−2x−15\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}-2 x-15}limx→−3​x2−2x−15x2−9​. Options: (A) 0 (B) 35\frac{3}{5}53​ (C) 34\frac{3}{4}43​ (D) 1 (E) nonexistent.

Problem 1117

Find lim⁡x→0tan⁡(2x)6xsec⁡(3x)\lim _{x \rightarrow 0} \frac{\tan (2 x)}{6 x \sec (3 x)}limx→0​6xsec(3x)tan(2x)​. Choose from (A) 0, (B) 16\frac{1}{6}61​, (C) 13\frac{1}{3}31​, (D) nonexistent.

Problem 1118

Find lim⁡x→2f(x)\lim _{x \rightarrow 2} f(x)limx→2​f(x) for the function f(x)=x2−4x2+x−6f(x)=\frac{x^{2}-4}{x^{2}+x-6}f(x)=x2+x−6x2−4​. Choices: (A) 0, (B) 23\frac{2}{3}32​, (C) 45\frac{4}{5}54​, (D) nonexistent.

Problem 1119

Find the value of kkk such that lim⁡x→2f(x)=3\lim _{x \rightarrow 2} f(x)=3limx→2​f(x)=3 for f(x)=(x−2)(x2−k2)(x2−4)(x−k)f(x)=\frac{(x-2)(x^{2}-k^{2})}{(x^{2}-4)(x-k)}f(x)=(x2−4)(x−k)(x−2)(x2−k2)​.

Problem 1120

Find lim⁡x→1f(x)\lim _{x \rightarrow 1} f(x)limx→1​f(x) for f(x)=x2−1x−1f(x)=\frac{x^{2}-1}{\sqrt{x}-1}f(x)=x​−1x2−1​. Choices: (A) 4, (B) 2, (C) 0, (D) nonexistent.

Problem 1121

Find lim⁡x→π4g(x)\lim _{x \rightarrow \frac{\pi}{4}} g(x)limx→4π​​g(x) for g(x)=cos⁡x−sin⁡x1−2sin⁡2xg(x)=\frac{\cos x-\sin x}{1-2 \sin ^{2} x}g(x)=1−2sin2xcosx−sinx​. Options: (A) 0 (B) 12\frac{1}{\sqrt{2}}2​1​ (C) 2\sqrt{2}2​ (D) Limit does not exist.

Problem 1122

Find the derivative of y=1xy=\frac{1}{x}y=x1​, calculate lim⁡x→∞−4x7x−3\lim_{x \to \infty} \frac{-4x}{7x-3}limx→∞​7x−3−4x​, and find two numbers with a difference of 4 that minimize their product.

Problem 1123

Find the average rate of change of f(x)=sin⁡xf(x)=\sin xf(x)=sinx from x1=π4x_{1}=\frac{\pi}{4}x1​=4π​ to x2=3π2x_{2}=\frac{3\pi}{2}x2​=23π​.

Problem 1124

Differentiate the following using the Product Rule: 1. y=x2(x+1)y=x^{2}(x+1)y=x2(x+1) 2. y=(x2+3)(x+6)y=(x^{2}+3)(x+6)y=(x2+3)(x+6) 3. y=x(x3+6)y=\sqrt{x}(x^{3}+6)y=x​(x3+6) 4. y=(2x2+4x−3)(3x+4)y=(2 x^{2}+4 x-3)(3 x+4)y=(2x2+4x−3)(3x+4)

Problem 1125

Differentiate these functions using the quotient rule: y=x+2x+3y=\frac{x+2}{x+3}y=x+3x+2​, y=1x(2x+1)y=\frac{1}{x(2x+1)}y=x(2x+1)1​, y=4x+13x+8y=\frac{4x+1}{3x+8}y=3x+84x+1​, y=2−3x1+xy=\frac{2-3x}{1+x}y=1+x2−3x​.

Problem 1126

Find the limit as xxx approaches 2 for the expression x2−4x+3x^{2} - 4x + 3x2−4x+3.

Problem 1127

Find the limit as xxx approaches 3 for the expression 4x+2x+4\frac{4x+2}{x+4}x+44x+2​.

Problem 1128

Find the limit: lim⁡x→π3sin⁡2xsin⁡x\lim _{x \rightarrow \frac{\pi}{3}} \frac{\sin 2 x}{\sin x}limx→3π​​sinxsin2x​.

Problem 1129

Calculate the limit: lim⁡x→0(1−cos⁡2x2x)\lim _{x \rightarrow 0}\left(\frac{\sqrt{1-\cos 2 x}}{\sqrt{2} x}\right)limx→0​(2​x1−cos2x​​).

Problem 1130

Determina la velocidad de una moneda en caída libre tras 2.0 s2.0 \mathrm{~s}2.0 s desde reposo en la Torre Colpatria.

Problem 1131

Calculate the difference quotient f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ for the function f(x)=−7x+6f(x)=-7x+6f(x)=−7x+6, where h≠0h \neq 0h=0.

Problem 1132

Problem 1133

Problem 1134

A golf ball is hit at 130 ft/s at 45∘45^{\circ}45∘. Find the distance to max height using h(x)=−32x21302+xh(x)=\frac{-32 x^{2}}{130^{2}}+xh(x)=1302−32x2​+x.

Problem 1135

The expression f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ is called the derivative of fff.

Problem 1136

Problem 1137

Find the rate of change of y=cos⁡5xy=\cos 5 xy=cos5x. Choices: A. sin⁡5x\sin 5 xsin5x, B. 5sin⁡5x5 \sin 5 x5sin5x, C. −sin⁡5x-\sin 5 x−sin5x, D. −5sin⁡5x-5 \sin 5 x−5sin5x.

Problem 1138

Find the derivative of f(x)=5x2f(x) = 5x^2f(x)=5x2.

Problem 1139

Find the integral of 1−x2x(1−2x)\frac{1-x^{2}}{x(1-2 x)}x(1−2x)1−x2​ and simplify to get the final expression.

Problem 1140

Calculate the integral ∫03(1−e−x) dx\int_{0}^{3}(1-e^{-x}) \, dx∫03​(1−e−x)dx with h=2h=2h=2.

Problem 1141

Calculate the integral ∫vtt⋅cos⁡(t5)⋅dt\int_{v}^{t} t \cdot \cos \left(\frac{t}{5}\right) \cdot d t∫vt​t⋅cos(5t​)⋅dt.

Problem 1142

Show that f(x)=x2−6x+1f(x)=x^{2}-6x+1f(x)=x2−6x+1 on [1,3] meets Lagrange's theorem. Find where the tangent is parallel to the line joining A(1,-4) and B(3,-8).

Problem 1143

Find the derivative of y=x2+3(4x2+2)4y=x^{2}+3(4x^{2}+2)^{4}y=x2+3(4x2+2)4.

Find the limit: $\lim _{x \rightarrow-\infty} \frac{x^{6}-3 x^{4}}{2 x^{2}-2 x+1}$ .

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{3x}{x+3}$ , with $h \neq 0$ . Simplify your answer.

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{10}{x^{2}}$ , where $h \neq 0$ . Simplify your answer.

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{11}{x^{2}}$ , simplifying your answer.

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{5 x}{x+6}$ , where $h \neq 0$ . Simplify your answer.

A rifle fires bullets with speed $v=(-5.35 \times 10^{7}) t^{2}+(2.30 \times 10^{5}) t$ . Find acceleration, position, time, speed, and barrel length.

Which table shows values of $g$ with $\lim_{x \to 7} g(x) = 6$ ? Options: (A), (B), (C), (D).

What does $\lim _{x \rightarrow 3} f(x)=5$ mean? Choose the correct interpretation: (A), (B), (C), or (D).

A car's distance is $s(t)$ in feet after $t$ seconds. Estimate its velocity at $t=6$ using options (A) to (D).

Find $\lim _{x \rightarrow 2^{+}} f(x)$ for the piecewise function: $f(x)=\{5x-3 \text{ if } x<2, 9 \text{ if } x=2, 4x+3 \text{ if } x>2\}$ .

Find $\lim _{x \rightarrow 2}(h(x)(5 f(x)+g(x)))$ given $f(2)=3$ , $g(2)=-6$ , $h(2)=-3$ , limits at $x=2$ .

Find $\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}-2 x-15}$ . Options: (A) 0 (B) $\frac{3}{5}$ (C) $\frac{3}{4}$ (D) 1 (E) nonexistent.

Find $\lim _{x \rightarrow 0} \frac{\tan (2 x)}{6 x \sec (3 x)}$ . Choose from (A) 0, (B) $\frac{1}{6}$ , (C) $\frac{1}{3}$ , (D) nonexistent.

Find $\lim _{x \rightarrow 2} f(x)$ for the function $f(x)=\frac{x^{2}-4}{x^{2}+x-6}$ . Choices: (A) 0, (B) $\frac{2}{3}$ , (C) $\frac{4}{5}$ , (D) nonexistent.

Find the value of $k$ such that $\lim _{x \rightarrow 2} f(x)=3$ for $f(x)=\frac{(x-2)(x^{2}-k^{2})}{(x^{2}-4)(x-k)}$ .

Find $\lim _{x \rightarrow 1} f(x)$ for $f(x)=\frac{x^{2}-1}{\sqrt{x}-1}$ . Choices: (A) 4, (B) 2, (C) 0, (D) nonexistent.

Find $\lim _{x \rightarrow \frac{\pi}{4}} g(x)$ for $g(x)=\frac{\cos x-\sin x}{1-2 \sin ^{2} x}$ . Options: (A) 0 (B) $\frac{1}{\sqrt{2}}$ (C) $\sqrt{2}$ (D) Limit does not exist.

Find the derivative of $y=\frac{1}{x}$ , calculate $\lim_{x \to \infty} \frac{-4x}{7x-3}$ , and find two numbers with a difference of 4 that minimize their product.

Find the average rate of change of $f(x)=\sin x$ from $x_{1}=\frac{\pi}{4}$ to $x_{2}=\frac{3\pi}{2}$ .

Differentiate the following using the Product Rule:
1. $y=x^{2}(x+1)$
2. $y=(x^{2}+3)(x+6)$
3. $y=\sqrt{x}(x^{3}+6)$
4. $y=(2 x^{2}+4 x-3)(3 x+4)$

Differentiate these functions using the quotient rule: $y=\frac{x+2}{x+3}$ , $y=\frac{1}{x(2x+1)}$ , $y=\frac{4x+1}{3x+8}$ , $y=\frac{2-3x}{1+x}$ .

Find the limit as $x$ approaches 2 for the expression $x^{2} - 4x + 3$ .

Find the limit as $x$ approaches 3 for the expression $\frac{4x+2}{x+4}$ .

Find the limit: $\lim _{x \rightarrow \frac{\pi}{3}} \frac{\sin 2 x}{\sin x}$ .

Calculate the limit: $\lim _{x \rightarrow 0}\left(\frac{\sqrt{1-\cos 2 x}}{\sqrt{2} x}\right)$ .

Determina la velocidad de una moneda en caída libre tras $2.0 \mathrm{~s}$ desde reposo en la Torre Colpatria.

Calculate the difference quotient $\frac{f(x+h)-f(x)}{h}$ for the function $f(x)=-7x+6$ , where $h \neq 0$ .

A golf ball is hit at 130 ft/s at $45^{\circ}$ . Find the distance to max height using $h(x)=\frac{-32 x^{2}}{130^{2}}+x$ .

The expression $\frac{f(x+h)-f(x)}{h}$ is called the derivative of $f$ .

Find the rate of change of $y=\cos 5 x$ . Choices: A. $\sin 5 x$ , B. $5 \sin 5 x$ , C. $-\sin 5 x$ , D. $-5 \sin 5 x$ .

Find the derivative of $f(x) = 5x^2$ .

Find the integral of $\frac{1-x^{2}}{x(1-2 x)}$ and simplify to get the final expression.

Calculate the integral $\int_{0}^{3}(1-e^{-x}) \, dx$ with $h=2$ .

Calculate the integral $\int_{v}^{t} t \cdot \cos \left(\frac{t}{5}\right) \cdot d t$ .

Show that $f(x)=x^{2}-6x+1$ on [1,3] meets Lagrange's theorem. Find where the tangent is parallel to the line joining A(1,-4) and B(3,-8).

Find the derivative of $y=x^{2}+3(4x^{2}+2)^{4}$ .

Find the derivatives of these functions: 1. $y=e^{-x}$ , 2. $y=e^{x+1}$ , 3. $y=\frac{1}{e^{5x}}$ .

Find the tangent line equation for the function $f(x)=e^{-x} \ln (x)$ at the point $(1,0)$ .

Find the average rate of change of $D(t)$ from $t=0.2$ to $t=0.4$ . Use units of miles per hour.

A cube's volume increases at $3 \frac{1}{2} \mathrm{~cm}^{3} \mathrm{~s}^{-1}$ . Find the base's change rate when length is $6 \mathrm{~cm}$ .

Find the derivative, integral, domain, range, and solution for the function $g(x)=\ln(2x-4)$ .

Evaluate the limit using the Squeeze Theorem: $\lim _{x \rightarrow 0} x^{2} \cos \left(\frac{1}{x^{2}}\right)$ .

Weekly cost to make $x$ cars and $y$ trucks is $C(x, y)=240,000+6,000x+4,000y$ . Find marginal costs and describe graphs.

Find all values of $a$ where $\lim_{x \to a} f(x)$ exists for the piecewise function $f(x)$ defined above.

Determine if the limits $L_{1} = \lim _{x \rightarrow 0+} \sin(\frac{\pi}{x})$ and $L_{2} = \lim _{x \rightarrow 0-} \sin(\frac{\pi}{x})$ exist.

Find the average velocity of a ball thrown upward on the moon, given $y(t)=25t-3t^2$ , over the interval from 1 to $1+h$ .

A ball is thrown with a velocity of $40 \mathrm{ft/s}$ . Find average velocity from $t=2$ for (a) 0.5s, 0.05s, 0.1s, 0.01s and (b) instantaneous velocity at $t=2$ .

Find the limit as $x$ approaches -1 for the expression $\frac{x^{2}+9 x+8}{8 x+8}$ .

Find the limit as $x$ approaches -8 for the expression $\frac{x^{2}-64}{4 x+32}$ .

Find the limit as $x$ approaches 5 for the expression $\frac{x^{2}-13 x+40}{x^{2}-8 x+15}$ .

Find $\lim _{x \rightarrow 0} \frac{3 x^{2}-\sin 2 x}{2 x-\sin 3 x}$ . What is the result? A) -1 B) 0 C) 2 D) Does not exist.

Find the limit: $\lim _{x \rightarrow 3^{-}} \frac{2-x}{x-3}$ . What is the result? A) $-\infty$ B) $+\infty$ C) 0 D) 1

Find the limit: $\lim _{x \rightarrow 0} \frac{4 \sin x}{2-2 \cos x}$ . What is the result? A) 2 B) $\pi$ C) 0 D) does not exist.

Find $\lim _{x \rightarrow c}[2 f(x)+3 g(x)]$ given $\lim _{x \rightarrow c} f(x)=\frac{-2}{3}$ and $\lim _{x \rightarrow c} g(x)=\frac{5}{4}$ .

Find the limit: $\lim _{x \rightarrow -5} \frac{x^{2}-25}{x^{2}-2x-35}$ .

Find the half-life of a substance decaying as $y = y_{0} e^{-0.019 t}$ . Round your answer to the nearest tenth.

Find the half-life of a substance decaying at $5.7\%$ per day. Round your answer to the nearest hundredth.

How long does it take for a bacteria population to double at a continuous growth rate of $1\%$ per hour? Round to the nearest hundredth.

Is the function $f(x) = \frac{x^{2}-x-20}{x^{2}-9 x+20}$ for $x \neq 5$ and $f(5) = 9$ continuous at $x=5$ ?

Calculate the integral: $\int \frac{2 x}{x^{2}-1} d x$

Find the difference quotient of $f(x)=x^{2}+4$ : calculate $\frac{f(x+h)-f(x)}{h}$ , $h \neq 0$ , and simplify.

Find the output level $x$ that minimizes the marginal cost $C(x)=x^{2}-140x+7100$ and determine the minimum cost.