2. Intro to Derivatives

In Exercises 83–88, find equations for the lines that are tangent, and the lines that are normal, to the curve at the given point.

x³/² + 2y³/² = 17, (1, 4)

Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.

f(x) = x³; P (1,1)

In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:

d. Find the equation for the tangent line to g at the point (f(x_0), x_0) located symmetrically across the 45° line y=x (which is the graph of the identity function). Use Theorem 1 to find the slope of this tangent line.

72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2

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

c. Find the slopes of the tangent lines to the graphs of h and k at (2, 2) and (−2, −2).

In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:

c. Find the equation for the tangent line to f at the specified point (x_0, f(x_0)).

72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2

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

ƒ(x) = x⁴

79–82. {Use of Tech} Visualizing tangent and normal lines

b. Graph the tangent and normal lines on the given graph.

(x²+y²)² = 25/3 (x²-y²); (x0,y0) = (2,-1) (lemniscate of Bernoulli)

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

Derivatives and tangent lines

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

f(x) = 1/3x-1; a= 2

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) = 1/3x-1; a= 2

In Exercises 11–18, find the slope of the function’s graph at the given point. Then find an equation for the line tangent to the graph there.

h(t) = t³ + 3t, (1, 4)

The population of the United States (in millions) by decade is given in the table, where t is the number of years after 1910. These data are plotted and fitted with a smooth curve y = p(t) in the figure. <IMAGE><IMAGE>

Compute the average rate of population growth from 1950 to 1960. 

Given the function $f(x)=x^2-10x+2$, calculate the slope of the tangent line at $x=2$.

Find the points on the curve y = tan x, -π/2 < x < π/2, where the normal line is parallel to the line y = -x/2. Sketch the curve and normal lines together, labeling each with its equation.