Chapter 4: Applications of Differential Calculus

# A. Slope; Critical Points

* If the derivative of *y* = *f* (*x*) exists at *P*(*x*1, *y*1), then the *slope* of the curve at *P* (which is defined to be the slope of the tangent to the curve at *P*) is *f*′(*x*1), the derivative of *f*(*x*) at *x* = *x*1.
* Any *c* in the domain of *f* such that either *f*′(*c*) = 0 or *f*′(*c*) is undefined is called a *critical point* or *critical value* of *f*. If *f* has a derivative everywhere, we find the critical points by solving the equation *f*′(*x*) = 0.

\
Example:


Solution:


\
### **Average and Instantaneous Rates of Change**

* If as *x* varies from *a* to *a* + *h*, the function *f* varies from *f*(*a*) to *f*(*a* + *h*), then we know that the difference quotient


is the average rate of change of *f* over the interval from *a* to *a* + *h*.
* The ***average velocity*** of a moving object over some time interval is the change in distance divided by the change in time.
* The ***average rate of growth*** of a colony of fruit flies over some interval of time is the change in size of the colony divided by the time elapsed.
* The ***average rate of change*** in the profit of a company on some gadget with respect to production is the change in profit divided by the change in the number of gadgets produced.

\
The (instantaneous) rate of change of *f* at *a*, or the derivative of *f* at *a*, is the limit of the average rate of change as *h* → 0:


* On the graph of *y* = *f*(*x*), the rate at which the *y*-coordinate changes with respect to the *x*-coordinate is *f*′(*x*), the **slope of the curve.**
* The rate at which *s*(*t*), the distance traveled by a particle in *t* seconds, changes with respect to time is *s*′(*t*), the **velocity** of the particle.
* The rate at which a manufacturer’s profit *P*(*x*) changes relative to the production level *x* is *P*′(*x*).

\
Example:

Let *G* = 400(15 − *t*)**²** be the number of gallons of water in a cistern *t* minutes after an outlet pipe is opened. Find the average rate of drainage during the first 5 minutes and the rate at which the water is running out at the end of 5 minutes.

\
Solution:


\
# B. Tangents to a Curve

* An *equation of the tangent* to the curve *y* = *f*(*x*) at point *P*(*x*1, *y*1) is

*y* − *y*1 = *f*′(*x*1)(*x* − *x*1)
* If the tangent to a curve is horizontal at a point, then the derivative at the point is 0.
* If the tangent is vertical at a point, then the derivative does not exist at the point.

**Example:**

Find an equation of the tangent to the curve of *f*(*x*) = *x*³ − 3*x*² at the point (1,−2).

SOLUTION:

Since *f*′(*x*) = 3*x*² − 6*x* and *f*′(1) = −3, an equation of the tangent is

*y* + 2 = −3(*x* − 1) or *y* + 3*x* = 1

\
# C. Increasing and Decreasing Functions

### Functions with Continuous Derivatives

A function *y* = *f*(*x*) is said to be *increasing/decreasing* on an interval if for all *a* and *b* in the interval such that *a*