?Derivatives and tangent linesa. For the following functions and points, find f’(a).b

Graph of the derivative of the sine curve a. Use the graph of y = sin x (see figure) to sketch the graph of the derivative of the sine function. b. Based upon your graph in part (a), what function equals \(\frac{d}{dx}(\sin x)\)?

Determining the unknown constant Let \(f(x)=\left\{\begin{array}{ll} 2 x^{2} & \text { if } x \leq 1 \\ a x-2 & \text { if } x>1 \end{array}\right.\) Determine a value of a (if possible) for which f’(1) exists.

Match the functions (a)-(d) in the first set of figures with the derivative functions (A)-(D) in the next set of figures.

A derivative formula a. Use the definition of the derivative to determine \(\frac{d}{dx}(ax^2+bx+c)\), where a, b, and c are constants. b. Use the result of part (a) to find \(\frac{d}{dx}(4x^2-3x+10)\).

A derivative formula a. Use the definition of the derivative to determine \(\frac{d}{dx}(\sqrt{ax+b})\), where a and b are constants. b. Use the result of part (a) to find \(\frac{d}{dx}(\sqrt{5x+9})\).

Evaluate the derivative of the following functions at the given point. y =1/(t+1); t = 1

Evaluate the derivative of the following functions at the given point. \(y=t-t^2;\ t=2\)

Evaluate the derivative of the following functions at the given point. \(c=2\sqrt{s}-1;\ s=25\)

Evaluate the derivative of the following functions at the given point. \(A=\pi r^2;\ r=3\)

Use the graph of f to sketch a graph of f’.

Use the graph of f to sketch a graph of f’.

Reproduce the graph of f and then sketch a graph of f’ on the same axes.

Reproduce the graph of f and then sketch a graph of f’ on the same axes.

Reproduce the graph of f and then sketch a graph of f’ on the same axes.

Determine whether the following statements are true and give an explanation or counterexample. a. For linear functions, the slope of any secant line always equals the slope of any tangent line. b. The slope of the secant line passing through the points P and Q is less than the slope of the tangent line at P. c. Consider the graph of the parabola \(f(x)=x^2\). For x > 0 and h > 0, the secant line through (x, f(x)) and (x + h, f(x + h)) always has a greater slope than the tangent line at (x, f(x)). d. If the function f is differentiable for all values of x, then f is continuous for all values of x.

Consider the line f(x) = mx + b, where m and b are constants. Show that f’(x) = m for all x. Interpret this result.

Calculating derivatives a. For the following functions, find f’ using the definition \(f'(x)=\lim _{h \rightarrow 0}\ \frac{f(x+h)-f(x)}{h}\) b. Determine an equation of the line tangent to the graph of f at (a, f(a)) for the given value of a. \(f(x)=\sqrt{3x-1};\ \ a=8\)

Calculating derivatives a. For the following functions, find f’ using the definition \(f'(x)=\lim _{h \rightarrow 0}\ \frac{f(x+h)-f(x)}{h}\) b. Determine an equation of the line tangent to the graph of f at (a, f(a)) for the given value of a. \(f(x)=\sqrt{x+2};\ \ a=7\)

Calculating derivatives a. For the following functions, find f’ using the definition \(f'(x)=\lim _{h \rightarrow 0}\ \frac{f(x+h)-f(x)}{h}\) b. Determine an equation of the line tangent to the graph of f at (a, f(a)) for the given value of a. \(f(x)=\frac{2}{3x+1};\ \ a=-1\)

Calculating derivatives a. For the following functions, find f’ using the definition \(f'(x)=\lim _{h \rightarrow 0}\ \frac{f(x+h)-f(x)}{h}\) b. Determine an equation of the line tangent to the graph of f at (a, f(a)) for the given value of a. \(f(x)=\frac{1}{x};\ \ a=-5\)

Use the points A, B, C, D, and E in the following graphs to answer these questions. a. At which point(s) is the slope of the curve negative? b. At which point(s) is the slope of the positive? c. Using A-E, list the slopes in decreasing order.

Use the points A, B, C, D, and E in the following graphs to answer these questions. a. At which point(s) is the slope of the curve negative? b. At which point(s) is the slope of the positive? c. Using A-E, list the slopes in decreasing order.

Finding f from f’ Sketch the graph of f’(x) = x (the derivative of f). Then, sketch a possible graph of f. Is there more than one possible graph?

Finding f from f’ Create the graph of a continuous function y = f(x), where \(f^{\prime}(x)=\left\{\begin{array}{cl} 1 & \text { if } x<0 \\ 0 & \text { if } 0<x<1 \\ -1 & \text { if } x>1 \end{array}\right.\) Is there more than one possible graph?

Energy is the capacity to do work and power is the rate at which energy is used or consumed. Therefore, if E(t) is the energy function for a system, then P(t) = E’(t) is the power function. A unit of energy is the kilowatt-hour (1 kWh is the amount of energy needed to light ten 100-W light bulbs for an hour); the corresponding units for power are kW. The following figure shows the energy consumed by a small community over a 25-hr period. a. Estimate the power at t = 10 and t = 20 hr. Be sure to include units in your calculations. b. At what times on the interval [0, 25] is the power zero? c. At what times on the interval [0, 25] is the power a maximum?

One-sided derivatives The left-hand and right-hand derivatives of a function at a point a are given by \(f'_{+}(a)=\lim _{h \rightarrow 0^+}\ \frac{f(a+h)-f(a)}{h}\) and \(f'_{-}(a)=\lim _{h \rightarrow 0^+}\ \frac{f(a+h)-f(a)}{h}\) provided these limits exist. The derivative f’(a) exists if and only if \(f'_{+}(a)=f'_{-}(a)\). a. Sketch the following functions. b. Compute \(f'_{+}(a)\) and \(f'_{-}(a)\) at the given point a. c. Is f continuous at a? Is f differentiable at a? f(x) = | x - 2 |; a = 2

One-sided derivatives The left-hand and right-hand derivatives of a function at a point a are given by \(f'_{+}(a)=\lim _{h \rightarrow 0^+}\ \frac{f(a+h)-f(a)}{h}\) and \(f'_{-}(a)=\lim _{h \rightarrow 0^+}\ \frac{f(a+h)-f(a)}{h}\) provided these limits exist. The derivative f’(a) exists if and only if \(f'_{+}(a)=f'_{-}(a)\). a. Sketch the following functions. b. Compute \(f'_{+}(a)\) and \(f'_{-}(a)\) at the given point a. c. Is f continuous at a? Is f differentiable at a? \(f(x)=\left\{\begin{array}{ll} 4-x^{2} & \text { if } x \leq 1 \\ 2 x+1 & \text { if } x>1 \end{array} ; a=1\right.\)

If a function f is continuous at a and \(\lim _{x \rightarrow a}\ |f'(x)|=\infty\), 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 59-60) is used. Use this definition to answer the following questions. The preceding definition of vertical tangent line includes four cases: \(\lim _{x \rightarrow a^+}\ f'(x)=-\infty\) and \(\lim _{x \rightarrow a^-}\ f'(x)=\infty\). Make a rough sketch of a (continuous) function that has a vertical tangent line at x = a in each of the four cases.

If a function f is continuous at a and \(\lim _{x \rightarrow a}\ |f'(x)|=\infty\), 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 59-60) is used. Use this definition to answer the following questions. Graph the following functions and determine the location of the vertical tangent lines. a. \(f(x)=(x-2)^{1/3}\) b. \(f(x)=(x+1)^{2/3}\) c. \(f(x)=\sqrt{|x-4|}\) d. \(f(x)=x^{5/3}-2x^{1/3}\)

If a function f is continuous at a and \(\lim _{x \rightarrow a}\ |f'(x)|=\infty\), 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 59-60) is used. Use this definition to answer the following questions. Verify that \(f(x)=x^{1/3}\) has a vertical tangent line at x =0.

The following limits represent the slope of a curve y = f(x) at the point (a, f(x)). Determine a function f and a number a; then, calculate the limit. \(\lim _{x \rightarrow 2}\ \frac{\frac{1}{x+1}-\frac{1}{3}}{x-2}\)

If a function f is continuous at a and \(\lim _{x \rightarrow a}\ |f'(x)|=\infty\), 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 59-60) is used. Use this definition to answer the following questions. Graph the following curves and determine the location of any vertical tangent lines. a. \(x^2+y^2=9\) b. \(x^2+y^2+2x=0\)

The following limits represent the slope of a curve y = f(x) at the point (a, f(x)). Determine a function f and a number a; then, calculate the limit. \(\lim _{h \rightarrow 0}\ \frac{\sqrt{2+h}-\sqrt{2}}{h}\)

The following limits represent the slope of a curve y = f(x) at the point (a, f(x)). Determine a function f and a number a; then, calculate the limit. \(\lim _{h \rightarrow 0}\ \frac{(2+h)^4-16}{h}\)

Is it differentiable? Is \(f(x)=\frac{x^2-5x+6}{x-2}\) differentiable at x = 2? Justify your answer.

The following limits represent the slope of a curve y = f(x) at the point (a, f(x)). Determine a function f and a number a; then, calculate the limit. \(\lim _{x \rightarrow 1}\ \frac{3x^2+4x-7}{x-1}\)

Derivative of x’’ Use the symbolic capabilities of a calculator to calculate f’(x) using the definition \(\lim _{h \rightarrow 0}\ \frac{f(x+h)-f(x)}{h}\) for the following functions. a. \(f(x) = x^2\) b. \(f(x) = x^3\) c. \(f(x) = x^4\) d. \(f(x) = x^100\) e. Based upon your answers to parts (a)-(d), propose a formula for f’(x) if \(f(x) = x^n\) where n is a positive integer.

Use definition (1) for the slope of a tangent line to explain how slopes of secant lines approach the slope of the tangent line at a point.

Explain why the slope of a secant line can be interpreted as an average rate of change.

Explain why the slope of the tangent line can be interpreted as an instantaneous rate of change.

Given the function f, what does f’ represent?

Given a function f and a point a in its domain, what does f’(a) represent?

Explain the relationships among the slope of a tangent line, the instantaneous rate of change, and the value of the derivative at a point.

Why is the notation \(\frac{dy}{dx}\) used to represent the derivative?

If f is differentiable at x = a, must f be continuous at x = a?

If f is continuous at x = a, must f be differentiable at x = a?

Give three different notations for the derivative of f with respect to x.

Equations of tangent lines by definition (1) a. Use definition (1) (p. 122) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. c. Plot the graph of f and the tangent line at P. \(f(x)=x^2-5;\ \ P(3,\ 4)\)

Equations of tangent lines by definition (1) a. Use definition (1) (p. 122) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. c. Plot the graph of f and the tangent line at P. \(f(x)=-3x^2-5x+1;\ \ P(1,\ -7)\)

Equations of tangent lines by definition (1) a. Use definition (1) (p. 122) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. c. Plot the graph of f and the tangent line at P. \(f(x)=-5x+1;\ \ P(1, \-4)\)

Equations of tangent lines by definition (1) a. Use definition (1) (p. 122) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. c. Plot the graph of f and the tangent line at P. \(f(x)=5;\ \ P(1,\ 5)\)

Equations of tangent lines by definition (1) a. Use definition (1) (p. 122) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. c. Plot the graph of f and the tangent line at P. \(f(x)=\frac{1}{x};\ \ P(-1,\ -1)\)

Equations of tangent lines by definition (1) a. Use definition (1) (p. 122) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. c. Plot the graph of f and the tangent line at P. \(f(x)=\frac{4}{x^2};\ \ P(-1,\ 4)\)

Equations of tangent lines by definition (2) a. Use definition (2) (p. 123) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. f(x) = 2x + 1; P(0, 1)

Equations of tangent lines by definition (2) a. Use definition (2) (p. 123) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. \(f(x)=3x^2-4x;\ \ P(1,\ -1)\)

Equations of tangent lines by definition (2) a. Use definition (2) (p. 123) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. \(f(x)=x^4;\ \ P(-1,\ 1)\)

Equations of tangent lines by definition (2) a. Use definition (2) (p. 123) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. \(f(x)=\frac{1}{2x+1};\ \ P(0,\ 1)\)

Equations of tangent lines by definition (2) a. Use definition (2) (p. 123) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. \(f(x)=\frac{1}{3-2x};\ \ P(-1,\ \frac{1}{5})\)

Equations of tangent lines by definition (2) a. Use definition (2) (p. 123) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. \(f(x)=\sqrt{x-1};\ \ P(2,\ 1)\)

Derivatives and tangent lines a. For the following functions and points, find f’(a). b. Determine an equation of the line tangent to the graph of f at (a, f(a)) for the given value of a. f(x) = 8x; a = - 3

Derivatives and tangent lines a. For the following functions and points, find f’(a). b. Determine an equation of the line tangent to the graph of f at (a, f(a)) for the given value of a. \(f(x)=x^2;\ \ a=3\)

Derivatives and tangent lines a. For the following functions and points, find f’(a). b. Determine an equation of the line tangent to the graph of f at (a, f(a)) for the given value of a. \(f(x)=4x^2+2x;\ \ a=-2\)

Derivatives and tangent lines a. For the following functions and points, find f’(a). b. Determine an equation of the line tangent to the graph of f at (a, f(a)) for the given value of a. \(f(x)=2x^3;\ \ a=10\)

Derivatives and tangent lines a. For the following functions and points, find f’(a). b. Determine an equation of the line tangent to the graph of f at (a, f(a)) for the given value of a. \(f(x)=\frac{1}{\sqrt{x}};\ \ a=¼\)

Derivatives and tangent lines a. For the following functions and points, find f’(a). b. Determine an equation of the line tangent to the graph of f at (a, f(a)) for the given value of a. \(f(x)=\frac{1}{x^2};\ \ a=1\)

Lines tangent to parabolas a. Find the derivative function f’ for the following functions f. b. Find an equation of the line tangent to the graph of f at (a, f(a)) for the given value of a. c. Graph f and the tangent line. \(f(x)=3x^2+2x-10;\ \ a=1\)

Lines tangent to parabolas a. Find the derivative function f’ for the following functions f. b. Find an equation of the line tangent to the graph of f at (a, f(a)) for the given value of a. c. Graph f and the tangent line. \(f(x)=3x^2;\ \ a=0\)

Lines tangent to parabolas a. Find the derivative function f’ for the following functions f. b. Find an equation of the line tangent to the graph of f at (a, f(a)) for the given value of a. c. Graph f and the tangent line. \(f(x)=5x^2-6x+1;\ \ a=2\)

Lines tangent to parabolas a. Find the derivative function f’ for the following functions f. b. Find an equation of the line tangent to the graph of f at (a, f(a)) for the given value of a. c. Graph f and the tangent line. \(f(x)=1-x^2;\ \ a=-1\)

Where is the function continuous? Differentiable? Use the graph of f in the figure to do the following. a. Find the values of x in (0, 3) at which f is not continuous. b. Find the values of x in (0, 3) at which f is not differentiable. c. Sketch a graph of f’.

Where is the function continuous? Differentiable? Use the graph of g in the figure to do the following. a. Find the values of x in (0, 4) at which g is not continuous. b. Find the values of x in (0, 4) at which g is not differentiable. c. Sketch a graph of g’.

Let p(t) represent the population of the Las Vegas metropolitan area t years after 1950, as shown in the table and figure. a. Compute the average rate of growth of Las Vegas from 1970 to 1980. b. Explain why the average rate of growth calculated in part (a) is a good estimate of the instantaneous rate of growth of Las Vegas in 1975. c. Compute the average rate of growth of Las Vegas from 1990 to 2000. Is this average rate of growth an overestimate or underestimate of the instantaneous rate of growth of Las Vegas in 2000? Approximate the growth rate in 2000.