2. Intro to Derivatives

An equation of the line tangent to the graph of g at x = 3 is y = 5x + 4. Find g(3) and g′(3).

If h(1) = 2 and h′(1) = 3, find an equation of the line tangent to the graph of h at x = 1.

If f′(−2) = 7, find an equation of the line tangent to the graph of f at the point (−2,4).

A projectile is fired vertically upward into the air; its position (in feet) above the ground after t seconds is given by the function s (t). For the following functions, use limits to determine the instantaneous velocity of the projectile at t = a seconds for the given value of a.

s(t) = -16t² + 128t + 192; a = 2

Use definition (1) (p. 133) to find the slope of the line tangent to the graph of f at P.

f(x) = x² - 5; P(3,4)

Use definition (1) (p. 133) to find the slope of the line tangent to the graph of f at P.

f(x) = -3x² - 5x + 1; P(1,-7)

Use definition (1) (p. 133) to find the slope of the line tangent to the graph of f at P.

f(x) = 1/x; P(-1,-1)

Use definition (1) (p. 133) to find the slope of the line tangent to the graph of f at P.

f(x) = 4/x²; P(-1,4)

Use definition (1) (p. 133) to find the slope of the line tangent to the graph of f at P.

f(x) = √(3x + 3); P(2,3)

Use definition (1) (p. 133) to find the slope of the line tangent to the graph of f at P.

f(x) = 2/√x; P(4,1)

Derivatives and tangent lines

b. Determine an equation of the line tangent to the graph of f at the point (a,f(a)) for the given value of a.

f(x) = x²; a=3

Derivatives and tangent lines

a. For the following functions and values of a, find f′(a).

f(x) = x²; a=3

Equations of tangent lines by definition (2)

b. Determine an equation of the tangent line at P.

f(x) = √x+3; P (1,2)

Simplify the difference quotient (ƒ(x)-ƒ(a)) / (x-a) for the following functions.

ƒ(x) = x⁴

Simplify the difference quotient (ƒ(x)-ƒ(a)) / (x-a) for the following functions.

ƒ(x) = (1/x) - x²