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‎posts/2025-12-01-gemini-api-multimodal.ipynb‎

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{
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"cell_type": "code",
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"execution_count": 64,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
3773-
"\n",
3774-
"================================================================================\n",
3775-
"Mathematical Problem Solving\n",
3776-
"================================================================================\n",
3777-
"\n",
3778-
"Calculus: Find the derivative of f(x) = (3x² + 2x - 1) * e^x...\n",
3779-
"--------------------------------------------------------------------------------\n",
3780-
"To find the derivative of $f(x) = (3x^2 + 2x - 1) * e^x$, we need to use the **Product Rule**.\n",
3781-
"\n",
3782-
"The Product Rule states that if $f(x) = u(x) * v(x)$, then its derivative $f'(x)$ is given by:\n",
3783-
"$f'(x) = u'(x) * v(x) + u(x) * v'(x)$\n",
3784-
"\n",
3785-
"Let's break down the function $f(x)$ into $u(x)$ and $v(x)$:\n",
3786-
"1. Let $u(x) = 3x^2 + 2x - 1$\n",
3787-
"2. Let $v(x) = e^x$\n",
3788-
"\n",
3789-
"Now, we need to find the derivatives of $u(x)$ and $v(x)$ separately.\n",
3790-
"\n",
3791-
"**Step 1: Find the derivative of u(x), denoted as u'(x).**\n",
3792-
"$u(x) = 3x^2 + 2x - 1$\n",
3793-
"To find $u'(x)$, we apply the Power Rule ($d/dx [x^n] = nx^{n-1}$) and the constant rule ($d/dx [c] = 0$) and the constant multiple rule ($d/dx [cx] = c$).\n",
3794-
"* The derivative of $3x^2$ is $3 * (2x^{2-1}) = 6x$.\n",
3795-
"* The derivative of $2x$ is $2 * (1x^{1-1}) = 2 * 1 = 2$.\n",
3796-
"* The derivative of $-1$ (a constant) is $0$.\n",
3797-
"\n",
3798-
"So, $u'(x) = 6x + 2 + 0$\n",
3799-
"$u'(x) = 6x + 2$\n",
3800-
"\n",
3801-
"**Step 2: Find the derivative of v(x), denoted as v'(x).**\n",
3802-
"$v(x) = e^x$\n",
3803-
"The derivative of $e^x$ is simply $e^x$.\n",
3804-
"\n",
3805-
"So, $v'(x) = e^x$\n",
3806-
"\n",
3807-
"**Step 3: Apply the Product Rule.**\n",
3808-
"Now, substitute $u(x)$, $u'(x)$, $v(x)$, and $v'(x)$ into the Product Rule formula:\n",
3809-
"$f'(x) = u'(x) * v(x) + u(x) * v'(x)$\n",
3810-
"$f'(x) = (6x + 2) * e^x + (3x^2 + 2x - 1) * e^x$\n",
3811-
"\n",
3812-
"**Step 4: Simplify the expression.**\n",
3813-
"Notice that $e^x$ is a common factor in both terms. We can factor it out:\n",
3814-
"$f'(x) = e^x * [(6x + 2) + (3x^2 + 2x - 1)]$\n",
3815-
"\n",
3816-
"Now, combine the like terms inside the square brackets:\n",
3817-
"$f'(x) = e^x * [3x^2 + (6x + 2x) + (2 - 1)]$\n",
3818-
"$f'(x) = e^x * [3x^2 + 8x + 1]$\n",
3819-
"\n",
3820-
"**Final Answer:**\n",
3821-
"The derivative of $f(x) = (3x^2 + 2x - 1) * e^x$ is:\n",
3822-
"$f'(x) = e^x (3x^2 + 8x + 1)$\n",
3823-
"\n",
3824-
"\n",
3825-
"Linear Algebra: Find the eigenvalues of the matrix:\n",
3826-
" [[2, 1...\n",
3827-
"--------------------------------------------------------------------------------\n",
3828-
"To find the eigenvalues of a matrix, we need to solve the characteristic equation, which is given by $\\det(A - \\lambda I) = 0$, where $A$ is the given matrix, $\\lambda$ represents the eigenvalues, and $I$ is the identity matrix of the same dimension as $A$.\n",
3829-
"\n",
3830-
"Given the matrix:\n",
3831-
"$A = \\begin{bmatrix} 2 & 1 \\\\ 1 & 2 \\end{bmatrix}$\n",
3832-
"\n",
3833-
"**Step 1: Form the matrix $(A - \\lambda I)$**\n",
3834-
"\n",
3835-
"First, we need to construct the matrix $(A - \\lambda I)$.\n",
3836-
"The identity matrix $I$ for a 2x2 matrix is $\\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix}$.\n",
3837-
"So, $\\lambda I = \\lambda \\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix} = \\begin{bmatrix} \\lambda & 0 \\\\ 0 & \\lambda \\end{bmatrix}$.\n",
3838-
"\n",
3839-
"Now, subtract $\\lambda I$ from $A$:\n",
3840-
"$A - \\lambda I = \\begin{bmatrix} 2 & 1 \\\\ 1 & 2 \\end{bmatrix} - \\begin{bmatrix} \\lambda & 0 \\\\ 0 & \\lambda \\end{bmatrix}$\n",
3841-
"$A - \\lambda I = \\begin{bmatrix} 2-\\lambda & 1 \\\\ 1 & 2-\\lambda \\end{bmatrix}$\n",
3842-
"\n",
3843-
"**Step 2: Calculate the determinant of $(A - \\lambda I)$**\n",
3844-
"\n",
3845-
"For a 2x2 matrix $\\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix}$, the determinant is $ad - bc$.\n",
3846-
"Applying this to $(A - \\lambda I)$:\n",
3847-
"$\\det(A - \\lambda I) = (2-\\lambda)(2-\\lambda) - (1)(1)$\n",
3848-
"$\\det(A - \\lambda I) = (2-\\lambda)^2 - 1$\n",
3849-
"\n",
3850-
"Expand the square term:\n",
3851-
"$(2-\\lambda)^2 = 4 - 4\\lambda + \\lambda^2$\n",
3852-
"\n",
3853-
"So, the determinant becomes:\n",
3854-
"$\\det(A - \\lambda I) = \\lambda^2 - 4\\lambda + 4 - 1$\n",
3855-
"$\\det(A - \\lambda I) = \\lambda^2 - 4\\lambda + 3$\n",
3856-
"\n",
3857-
"**Step 3: Solve the characteristic equation**\n",
3858-
"\n",
3859-
"Set the determinant equal to zero to find the eigenvalues:\n",
3860-
"$\\lambda^2 - 4\\lambda + 3 = 0$\n",
3861-
"\n",
3862-
"This is a quadratic equation. We can solve it by factoring, using the quadratic formula, or completing the square.\n",
3863-
"Let's factor the quadratic equation. We need two numbers that multiply to 3 and add up to -4. These numbers are -1 and -3.\n",
3864-
"So, we can factor the equation as:\n",
3865-
"$(\\lambda - 1)(\\lambda - 3) = 0$\n",
3866-
"\n",
3867-
"Setting each factor to zero gives the eigenvalues:\n",
3868-
"$\\lambda - 1 = 0 \\implies \\lambda_1 = 1$\n",
3869-
"$\\lambda - 3 = 0 \\implies \\lambda_2 = 3$\n",
3870-
"\n",
3871-
"**Conclusion:**\n",
3872-
"\n",
3873-
"The eigenvalues of the matrix $\\begin{bmatrix} 2 & 1 \\\\ 1 & 2 \\end{bmatrix}$ are $\\boxed{1 \\text{ and } 3}$.\n",
3874-
"\n",
3875-
"\n",
3876-
"Statistics: Given data: [12, 15, 18, 22, 25, 30, 35]\n",
3877-
" C...\n",
3878-
"--------------------------------------------------------------------------------\n",
3879-
"Let's calculate the mean, median, variance, and standard deviation for the given data step by step.\n",
3880-
"\n",
3881-
"Given data: $[12, 15, 18, 22, 25, 30, 35]$\n",
3882-
"\n",
3883-
"First, let's count the number of data points, $n$.\n",
3884-
"$n = 7$\n",
3885-
"\n",
3886-
"---\n",
3887-
"\n",
3888-
"### Step 1: Calculate the Mean ($\\bar{x}$)\n",
3889-
"\n",
3890-
"The mean is the sum of all data points divided by the number of data points.\n",
3891-
"\n",
3892-
"**Formula:** $\\bar{x} = \\frac{\\sum x}{n}$\n",
3893-
"\n",
3894-
"1. **Sum the data points ($\\sum x$):**\n",
3895-
" $\\sum x = 12 + 15 + 18 + 22 + 25 + 30 + 35 = 157$\n",
3896-
"\n",
3897-
"2. **Divide by the number of data points ($n$):**\n",
3898-
" $\\bar{x} = \\frac{157}{7}$\n",
3899-
" $\\bar{x} \\approx 22.42857...$\n",
3900-
"\n",
3901-
"**Mean ($\\bar{x}$) = 22.43** (rounded to two decimal places)\n",
3902-
"\n",
3903-
"---\n",
3904-
"\n",
3905-
"### Step 2: Calculate the Median\n",
3906-
"\n",
3907-
"The median is the middle value in a dataset when it is ordered from least to greatest.\n",
3908-
"\n",
3909-
"1. **Order the data:**\n",
3910-
" The given data is already ordered: $[12, 15, 18, 22, 25, 30, 35]$\n",
3911-
"\n",
3912-
"2. **Find the position of the median:**\n",
3913-
" Since $n=7$ (an odd number), the median is at the position $\\frac{n+1}{2}$.\n",
3914-
" Position = $\\frac{7+1}{2} = \\frac{8}{2} = 4^{th}$\n",
3915-
"\n",
3916-
"3. **Identify the value at that position:**\n",
3917-
" The 4th value in the ordered list is 22.\n",
3918-
"\n",
3919-
"**Median = 22**\n",
3920-
"\n",
3921-
"---\n",
3922-
"\n",
3923-
"### Step 3: Calculate the Variance ($s^2$)\n",
3924-
"\n",
3925-
"Variance measures how spread out the data points are from the mean. We will calculate the sample variance, which is commonly used when the data is a sample from a larger population.\n",
3926-
"\n",
3927-
"**Formula for Sample Variance:** $s^2 = \\frac{\\sum (x_i - \\bar{x})^2}{n-1}$\n",
3928-
"\n",
3929-
"We will use the more precise value for the mean: $\\bar{x} = \\frac{157}{7}$.\n",
3930-
"\n",
3931-
"1. **Calculate the difference of each data point from the mean ($x_i - \\bar{x}$):**\n",
3932-
" * $12 - \\frac{157}{7} = \\frac{84-157}{7} = -\\frac{73}{7}$\n",
3933-
" * $15 - \\frac{157}{7} = \\frac{105-157}{7} = -\\frac{52}{7}$\n",
3934-
" * $18 - \\frac{157}{7} = \\frac{126-157}{7} = -\\frac{31}{7}$\n",
3935-
" * $22 - \\frac{157}{7} = \\frac{154-157}{7} = -\\frac{3}{7}$\n",
3936-
" * $25 - \\frac{157}{7} = \\frac{175-157}{7} = \\frac{18}{7}$\n",
3937-
" * $30 - \\frac{157}{7} = \\frac{210-157}{7} = \\frac{53}{7}$\n",
3938-
" * $35 - \\frac{157}{7} = \\frac{245-157}{7} = \\frac{88}{7}$\n",
3939-
"\n",
3940-
"2. **Square each of these differences ($(x_i - \\bar{x})^2$):**\n",
3941-
" * $(-\\frac{73}{7})^2 = \\frac{5329}{49}$\n",
3942-
" * $(-\\frac{52}{7})^2 = \\frac{2704}{49}$\n",
3943-
" * $(-\\frac{31}{7})^2 = \\frac{961}{49}$\n",
3944-
" * $(-\\frac{3}{7})^2 = \\frac{9}{49}$\n",
3945-
" * $(\\frac{18}{7})^2 = \\frac{324}{49}$\n",
3946-
" * $(\\frac{53}{7})^2 = \\frac{2809}{49}$\n",
3947-
" * $(\\frac{88}{7})^2 = \\frac{7744}{49}$\n",
3948-
"\n",
3949-
"3. **Sum the squared differences ($\\sum (x_i - \\bar{x})^2$):**\n",
3950-
" $\\sum (x_i - \\bar{x})^2 = \\frac{5329 + 2704 + 961 + 9 + 324 + 2809 + 7744}{49} = \\frac{19880}{49}$\n",
3951-
"\n",
3952-
"4. **Divide by $(n-1)$:**\n",
3953-
" $n-1 = 7-1 = 6$\n",
3954-
" $s^2 = \\frac{\\frac{19880}{49}}{6} = \\frac{19880}{49 \\times 6} = \\frac{19880}{294}$\n",
3955-
" $s^2 \\approx 67.68707...$\n",
3956-
"\n",
3957-
"**Variance ($s^2$) = 67.69** (rounded to two decimal places)\n",
3958-
"\n",
3959-
"---\n",
3960-
"\n",
3961-
"### Step 4: Calculate the Standard Deviation ($s$)\n",
3962-
"\n",
3963-
"The standard deviation is the square root of the variance. It provides a measure of the typical distance of data points from the mean, in the original units of the data.\n",
3964-
"\n",
3965-
"**Formula for Sample Standard Deviation:** $s = \\sqrt{s^2}$\n",
3966-
"\n",
3967-
"1. **Take the square root of the variance:**\n",
3968-
" $s = \\sqrt{\\frac{19880}{294}}$\n",
3969-
" $s = \\sqrt{67.68707...}$\n",
3970-
" $s \\approx 8.22721...$\n",
3971-
"\n",
3972-
"**Standard Deviation ($s$) = 8.23** (rounded to two decimal places)\n",
3973-
"\n",
3974-
"---\n",
3975-
"\n",
3976-
"### Summary of Results:\n",
3977-
"\n",
3978-
"* **Mean ($\\bar{x}$): 22.43**\n",
3979-
"* **Median: 22**\n",
3980-
"* **Variance ($s^2$): 67.69**\n",
3981-
"* **Standard Deviation ($s$): 8.23**\n",
3982-
"\n",
3983-
"\n",
3984-
"Optimization: A rectangular garden has perimeter of 60m.\n",
3985-
" ...\n",
3986-
"--------------------------------------------------------------------------------\n",
3987-
"Here's a step-by-step solution to maximize the area of the rectangular garden:\n",
3988-
"\n",
3989-
"---\n",
3990-
"\n",
3991-
"### Step 1: Understand the Problem and Define Variables\n",
3992-
"\n",
3993-
"* **Goal:** Maximize the area of a rectangular garden.\n",
3994-
"* **Constraint:** The perimeter of the garden is fixed at 60 meters.\n",
3995-
"* **Unknowns:** The dimensions (length and width) of the rectangle.\n",
3996-
"\n",
3997-
"Let:\n",
3998-
"* `L` be the length of the garden (in meters)\n",
3999-
"* `W` be the width of the garden (in meters)\n",
4000-
"* `P` be the perimeter of the garden (in meters)\n",
4001-
"* `A` be the area of the garden (in square meters)\n",
4002-
"\n",
4003-
"---\n",
4004-
"\n",
4005-
"### Step 2: Formulate Equations Based on the Given Information\n",
4006-
"\n",
4007-
"We know the standard formulas for the perimeter and area of a rectangle:\n",
4008-
"\n",
4009-
"1. **Perimeter Formula:** $P = 2L + 2W$\n",
4010-
" * Given $P = 60m$, so:\n",
4011-
" $60 = 2L + 2W$\n",
4012-
"\n",
4013-
"2. **Area Formula:** $A = L \\times W$\n",
4014-
" * This is the function we want to maximize.\n",
4015-
"\n",
4016-
"---\n",
4017-
"\n",
4018-
"### Step 3: Express Area as a Function of a Single Variable\n",
4019-
"\n",
4020-
"To maximize the area, we need to express the area equation in terms of only one variable (either L or W). We can use the perimeter equation for this.\n",
4021-
"\n",
4022-
"From the perimeter equation:\n",
4023-
"$60 = 2L + 2W$\n",
4024-
"\n",
4025-
"Divide the entire equation by 2:\n",
4026-
"$30 = L + W$\n",
4027-
"\n",
4028-
"Now, solve for one variable (let's solve for L):\n",
4029-
"$L = 30 - W$\n",
4030-
"\n",
4031-
"Substitute this expression for `L` into the Area formula:\n",
4032-
"$A = (30 - W) \\times W$\n",
4033-
"$A(W) = 30W - W^2$\n",
4034-
"\n",
4035-
"Now we have the area `A` as a function of only the width `W`. This is a quadratic function, which graphs as a parabola opening downwards (due to the -W² term), meaning it has a maximum point.\n",
4036-
"\n",
4037-
"---\n",
4038-
"\n",
4039-
"### Step 4: Find the Maximum Area Using Calculus (Derivative)\n",
4040-
"\n",
4041-
"To find the value of `W` that maximizes `A(W)`, we take the first derivative of `A(W)` with respect to `W` and set it equal to zero.\n",
4042-
"\n",
4043-
"$A(W) = 30W - W^2$\n",
4044-
"\n",
4045-
"Find the first derivative $A'(W)$:\n",
4046-
"$A'(W) = \\frac{d}{dW}(30W - W^2)$\n",
4047-
"$A'(W) = 30 - 2W$\n",
4048-
"\n",
4049-
"Set the derivative to zero to find critical points (where the slope of the tangent line is zero, indicating a potential maximum or minimum):\n",
4050-
"$30 - 2W = 0$\n",
4051-
"\n",
4052-
"Solve for `W`:\n",
4053-
"$30 = 2W$\n",
4054-
"$W = \\frac{30}{2}$\n",
4055-
"$W = 15$ meters\n",
4056-
"\n",
4057-
"---\n",
4058-
"\n",
4059-
"### Step 5: Calculate the Other Dimension (Length)\n",
4060-
"\n",
4061-
"Now that we have the optimal width `W`, we can find the corresponding length `L` using the relationship we established in Step 3:\n",
4062-
"\n",
4063-
"$L = 30 - W$\n",
4064-
"$L = 30 - 15$\n",
4065-
"$L = 15$ meters\n",
4066-
"\n",
4067-
"---\n",
4068-
"\n",
4069-
"### Step 6: Verify the Maximum Area and State the Conclusion\n",
4070-
"\n",
4071-
"The dimensions that maximize the area are:\n",
4072-
"* Length `L` = 15 meters\n",
4073-
"* Width `W` = 15 meters\n",
4074-
"\n",
4075-
"Let's calculate the maximum area:\n",
4076-
"$A = L \\times W$\n",
4077-
"$A = 15 \\times 15$\n",
4078-
"$A = 225$ square meters\n",
4079-
"\n",
4080-
"We can also quickly check the perimeter: $P = 2(15) + 2(15) = 30 + 30 = 60m$, which matches the given constraint.\n",
4081-
"\n",
4082-
"**Conclusion:**\n",
4083-
"\n",
4084-
"The dimensions that maximize the area of a rectangular garden with a perimeter of 60m are **15 meters by 15 meters**. This means the garden should be a square. The maximum area achieved is **225 square meters**.\n",
4085-
"\n"
4086-
]
4087-
}
4088-
],
4089-
"source": [
4090-
"print(\"\\n\" + \"=\"*80)\n",
4091-
"print(\"Mathematical Problem Solving\")\n",
4092-
"print(\"=\"*80)\n",
4093-
"\n",
4094-
"math_problems = [\n",
4095-
" {\n",
4096-
" \"type\": \"Calculus\",\n",
4097-
" \"problem\": \"Find the derivative of f(x) = (3x² + 2x - 1) * e^x\"\n",
4098-
" },\n",
4099-
" {\n",
4100-
" \"type\": \"Linear Algebra\",\n",
4101-
" \"problem\": \"\"\"Find the eigenvalues of the matrix:\n",
4102-
" [[2, 1],\n",
4103-
" [1, 2]]\"\"\"\n",
4104-
" },\n",
4105-
" {\n",
4106-
" \"type\": \"Statistics\",\n",
4107-
" \"problem\": \"\"\"Given data: [12, 15, 18, 22, 25, 30, 35]\n",
4108-
" Calculate: mean, median, variance, and standard deviation\"\"\"\n",
4109-
" },\n",
4110-
" {\n",
4111-
" \"type\": \"Optimization\",\n",
4112-
" \"problem\": \"\"\"A rectangular garden has perimeter of 60m.\n",
4113-
" What dimensions maximize the area?\"\"\"\n",
4114-
" }\n",
4115-
"]\n",
4116-
"\n",
4117-
"for prob in math_problems:\n",
4118-
" prompt = f\"\"\"Solve this {prob['type']} problem step by step:\n",
4119-
"\n",
4120-
"{prob['problem']}\n",
4121-
"\n",
4122-
"Show all work and explain each step.\"\"\"\n",
4123-
"\n",
4124-
" response = client.models.generate_content(\n",
4125-
" model=\"gemini-2.0-flash-thinking-exp-1219\",\n",
4126-
" contents=prompt\n",
4127-
" )\n",
4128-
" print(f\"\\n{prob['type']}: {prob['problem'][:50]}...\")\n",
4129-
" print(\"-\" * 80)\n",
4130-
" print(response.text)\n",
4131-
" print()"
4132-
]
4133-
},
41343764
{
41353765
"cell_type": "markdown",
41363766
"metadata": {},

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