Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

2. Intro to Derivatives

Tangent Lines and Derivatives

Calculus

2. Intro to Derivatives

Tangent Lines and Derivatives

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Multiple Choice

Find an equation of the tangent plane to the surface $z = x^{2} + y^{2}$ at the point $(1, 2, 5)$ .

z = 2 (x - 1) + 4 (y - 2) + 5

z = 2 x + 4 y - 5

z = 2 x + 4 y - 1

z = 2 x + 4 y - 5

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Verified step by step guidance

Step 1: Recall the general formula for the equation of the tangent plane to a surface z = f(x, y) at a point (x₀, y₀, z₀). The formula is given by: z - z₀ = fₓ(x₀, y₀)(x - x₀) + fᵧ(x₀, y₀)(y - y₀), where fₓ and fᵧ are the partial derivatives of f with respect to x and y, respectively.

Step 2: Compute the partial derivative of the given surface z = x² + y² with respect to x. This gives fₓ(x, y) = ∂/∂x(x² + y²) = 2x.

Step 3: Compute the partial derivative of the given surface z = x² + y² with respect to y. This gives fᵧ(x, y) = ∂/∂y(x² + y²) = 2y.

Step 4: Evaluate the partial derivatives at the given point (1, 2). For fₓ(1, 2), substitute x = 1 and y = 2 into fₓ(x, y) = 2x, resulting in fₓ(1, 2) = 2(1) = 2. Similarly, for fᵧ(1, 2), substitute x = 1 and y = 2 into fᵧ(x, y) = 2y, resulting in fᵧ(1, 2) = 2(2) = 4.

Step 5: Substitute the values x₀ = 1, y₀ = 2, z₀ = 5, fₓ(1, 2) = 2, and fᵧ(1, 2) = 4 into the tangent plane formula z - z₀ = fₓ(x₀, y₀)(x - x₀) + fᵧ(x₀, y₀)(y - y₀). Simplify the equation to find the tangent plane: z = 2(x - 1) + 4(y - 2) + 5.