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

1. Limits and Continuity

Introduction to Limits

Calculus

1. Limits and Continuity

Introduction to Limits

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Multiple Choice

Suppose a contour map is given for a function $f$ , and you are asked to estimate the partial derivatives $f_{x} (2, 1)$ and $f_{y} (2, 1)$ at the point $(2, 1)$ . Which of the following best describes how you would use the contour map to estimate these partial derivatives?

Estimate

f_{x} (2, 1)

and

f_{y} (2, 1)

by finding the steepest ascent direction at

(2, 1)

and using that value for both partial derivatives.

Estimate

f_{x} (2, 1)

and

f_{y} (2, 1)

by counting the total number of contour lines that pass through

(2, 1)

Estimate

f_{x} (2, 1)

by observing the change in

f

as you move in the

x

-direction from

(2, 1)

, and estimate

f_{y} (2, 1)

by observing the change in

f

as you move in the

y

-direction from

(2, 1)

, using the spacing of the contour lines.

Estimate both

f_{x} (2, 1)

and

f_{y} (2, 1)

by measuring the distance between contour lines along any diagonal direction through

(2, 1)

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Verified step by step guidance

Step 1: Understand the concept of partial derivatives. Partial derivatives measure the rate of change of a function with respect to one variable while keeping the other variables constant. For f_x(2, 1), we focus on changes in the x-direction, and for f_y(2, 1), we focus on changes in the y-direction.

Step 2: Locate the point (2, 1) on the contour map. Contour maps represent levels of constant function values, and the spacing between contour lines indicates the rate of change of the function.

Step 3: To estimate f_x(2, 1), observe the change in the function value as you move horizontally (in the x-direction) from the point (2, 1). The closer the contour lines are in this direction, the steeper the rate of change, and the larger the magnitude of f_x(2, 1).

Step 4: To estimate f_y(2, 1), observe the change in the function value as you move vertically (in the y-direction) from the point (2, 1). Similar to the x-direction, the spacing of contour lines in the y-direction determines the magnitude of f_y(2, 1).

Step 5: Use the spacing of the contour lines to approximate the values of f_x(2, 1) and f_y(2, 1). If the contour lines are closer together, the derivative will have a larger magnitude, and if they are farther apart, the derivative will have a smaller magnitude.