3. Techniques of Differentiation

{Use of Tech} Equations of tangent lines

Find an equation of the line tangent to the given curve at a.

y = e^x; a = ln 3

Let f(x) = x² - 6x + 5.

Find the values of x for which the slope of the curve y = f(x) is 0.

Given that f'(3) = 6 and g'(3) = -2 find (f+g)'(3).

Let f(x) = x² - 6x + 5.

Find the values of x for which the slope of the curve y = f(x) is 2.

Let f(x) = 4√x - x.

Find all points on the graph of f at which the tangent line is horizontal.

Let f(x) = 4√x - x.

Find all points on the graph of f at which the tangent line has slope -1/2.

Suppose f(3) = 1 and f′(3) = 4. Let g(x) = x² + f(x) and h(x) = 3f(x).

Find an equation of the line tangent to y = g(x) at x = 3.

Suppose f(3) = 1 and f′(3) = 4. Let g(x) = x² + f(x) and h(x) = 3f(x).

Find an equation of the line tangent to y = h(x) at x = 3.

Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.

y = x⁵

The following limits represent f'(a) for some function f and some real number a.

Find a possible function f and number a.

lim x🠂0 e^x-1 / x

Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.

f(x) = 5x³

The following limits represent f'(a) for some function f and some real number a.

b. Evaluate the limit by computing f'(a).

lim x🠂0 e^x-1 / x

Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.

h(t) = t²/2 + 1

The following limits represent f'(a) for some function f and some real number a.

b. Evaluate the limit by computing f'(a).

lim x🠂1 x¹⁰⁰-1 / x-1

Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.

f(v) = v¹⁰⁰+e^v+10