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 line to the curve $y = e^{x} cos (x) + sin (x)$ at the point $(0, 1)$ .

y = x

y = 2 x + 1

y = x + 1

y = - x + 1

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Verified step by step guidance

Step 1: Recall that the equation of a tangent line to a curve at a given point is given by y = f'(x₀)(x - x₀) + f(x₀), where f'(x₀) is the derivative of the function evaluated at x₀, and f(x₀) is the value of the function at x₀.

Step 2: Start by finding the derivative of the given function y = e^x * cos(x) + sin(x). Use the product rule for the term e^x * cos(x), which states (uv)' = u'v + uv', and the derivative of sin(x), which is cos(x).

Step 3: Evaluate the derivative at the given point x = 0. Substitute x = 0 into the derivative expression to find the slope of the tangent line at this point.

Step 4: Evaluate the original function y = e^x * cos(x) + sin(x) at x = 0 to find the y-coordinate of the point of tangency. This gives the point (x₀, y₀) = (0, 1).

Step 5: Substitute the slope from Step 3 and the point (0, 1) into the tangent line equation y = f'(x₀)(x - x₀) + f(x₀) to find the equation of the tangent line.