Q2Q1 Bascal - Definition of Derivatives

Given the function f(x)=x^2+3x−2 at the point x=1, what is the equation of the tangent line?

• y = 2x -1

• y = 4x -3

• y = 5x - 3

• y = x - 1

At which point does the function g(x)=x^4 − 4x^3 + 6x^2 - 4x + 1 have a horizontal tangent line?

• (1, 0)

• (0, 1)

• (2, 1)

• (0, 0)

In the context of finding tangent lines, what does the limit of the difference quotient represent?

• The slope of the tangent line

• The area under the curve

• The arc length of the curve

• The curvature of the curve

For a given function f(x), how is the slope of the tangent line at a point 'a' calculated using limits?

• Taking the derivative of f(x) at x = a

• Evaluating the function at x = a

• Taking the limit of the difference quotient as h approaches zero

• Using the fundamental theorem of calculus

When finding the equation of the normal line to a curve at a specific point, what role does the slope of the tangent line play?

• It represents the x-coordinate of the point

• It represents the y-coordinate of the point

• It is perpendicular to the normal line

• It determines the curvature of the curve

What happens to the slope of the secant line as the two points on the curve come closer together?

• It becomes steeper

• It becomes flatter

• It remains constant

• It approaches the slope of the tangent line

Consider the function f(x) = x^2.
What does the limit expression lim(h->0) [f(x + h) - f(x)] / h represent?

• The slope of the tangent line at the point x on the curve.

• The integral of the function f(x) from x to x + h.

• The area under the curve of the function f(x).

• The derivative of the function f(x) with respect to x.

Consider the function f(x) = x^3 - 2x^2 + 5x - 3. What is the derivative of f(x) at x = 2 using the limit definition of derivatives?

• 5

• 9

• 13

• 19

A function g(x) is defined as g(x) = √x. What is the slope of the tangent line to g(x) at x = 4 using the limit definition of derivatives?

• 1/2

• 1/4

• 1/8

• 1/16

For the function h(x) = sin(x), what is the derivative of h(x) at x = π/4 using the limit definition of derivatives?

• 1

• √2/2

• √3/2

• π/4