2. Intro to Derivatives

Use the results of Exercise 55 to show that the functions in Exercises 56–60 have inverses over their domains. Find a formula for df⁻¹/dx using Theorem 1.

f(x) = (1 − x)³

Find an equation of the tangent line to the curve $y=e^{x}cos\left(x\right)+sin\left(x\right)$ at the point $(0,1)$.

Vertical tangent lines If a function f is continuous at a and lim x→a| f′(x)|=∞, then the curve y=f(x) has a vertical tangent line at a, and the equation of the tangent line is x=a. If a is an endpoint of a domain, then the appropriate one-sided derivative (Exercises 71–72) is used. Use this information to answer the following questions.

Graph the following curves and determine the location of any vertical tangent lines.

a. x²+y² = 9

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

f(x) = 1/x; a= -5

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 the definition of the derivative to evaluate the following limits.

${\lim }_{h\to 0}{\frac{\ln (e^8+h)-8}{h}}_{}$

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

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.

g(x) = 8 / x², (2, 2)

Derivatives and tangent lines

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

f(x) = 8x; a = −3

Find an equation of the tangent plane to the surface $z=x^2+y^2$ at the point $(1,2,5)$.

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)

Given the parametric equations $x=t^2-4$ and $y=t^2-2t$, find the equation of the tangent line to the curve at the point where $t=3$.

Find an equation of the line tangent to the following curves at the given value of x.

y = 4 sin x cos x; x = π/3

Find an equation of the tangent line to the curve $y=e^x$ at the point $(1,e)$.

What is the slope of the tangent line to the curve $y^3-x{y}^2+x^3=5$ at the point $(1,2)$?