Chapter 13

Let \(f(x)=2 x+1, x_{1}=0, x_{2}=2, x_{3}=4, x_{4}=6,\) and \(\Delta x=2\) (a) Find \(\sum_{i=1}^{4} f\left(x_{i}\right) \Delta x\) (b) The sum in part (a) approximates a definite integral by using rectangles. The height of each rectangle is given by the value of the function at the left endpoint. Write the definite integral that the sum approximates.

Determine each limit. Refer to the accompanying graph of \(y=f(x)\) when it is given. Do not use a calculator. (a) \(\lim _{x \rightarrow 2^{+}} f(x)\) (b) \(\lim _{x \rightarrow 2^{-}} f(x)\) where \(f(x)=\left\\{\begin{array}{ll}x & \text { if } x<2 \\ 3 & \text { if } x=2 \\ 4 & \text { if } x>2\end{array}\right.\) (GRAPH CANNOT COPY).

Let \(\lim _{x \rightarrow 4} f(x)=16\) and \(\lim _{x \rightarrow 4} g(x)=8 .\) Use the limit rules to find each limit. Do not use a calculator. $$\lim _{x \rightarrow 4}[f(x)-g(x)]$$

Tell whether each statement is true or false. If a function \(f\) is defined at \(x=a,\) then \(\lim f(x)\) is always equal to \(f(a)\)

Use the midpoint rule with \(n=4\) to approximate the area above the \(x\) -axis bounded by the graph of $$f(x)=\sqrt{16-x^{2}}$$ in the first quadrant.

Let \(\lim _{x \rightarrow 4} f(x)=16\) and \(\lim _{x \rightarrow 4} g(x)=8 .\) Use the limit rules to find each limit. Do not use a calculator. $$\lim _{x \rightarrow 4}[g(x) \cdot f(x)]$$

Tell whether each statement is true or false. If \(\lim f(x)\) does not exist, then \(f(x)\) necessarily approaches one value as \(x\) approaches \(a\) from the left and a different value as \(x\) approaches \(a\) from the right.

Approximate the area under the graph of \(f(x)\) and above the \(x\) -axis, using each of the following methods with \(n=4\). (a) Use left endpoints. (b) Use right endpoints. (c) Average the answers in parts ( \(a\) ) and ( \(b\) ). (d) Use midpoints. \(f(x)=3 x+2\) from \(x=1\) to \(x=5\)

Determine each limit. Refer to the accompanying graph of \(y=f(x)\) when it is given. Do not use a calculator. (a) \(\lim _{x \rightarrow 3^{+}} f(x)\) (b) \(\lim _{x \rightarrow 3^{-}} f(x)\) where \(f(x)=\frac{x}{5(3-x)^{3}}\) (GRAPH CANNOT COPY).

Let \(\lim _{x \rightarrow 4} f(x)=16\) and \(\lim _{x \rightarrow 4} g(x)=8 .\) Use the limit rules to find each limit. Do not use a calculator. $$\lim _{x \rightarrow 4} \frac{f(x)}{g(x)}$$

Tell whether each statement is true or false. If \(\lim _{x \rightarrow 1} f(x)=5,\) then 1 must be in the domain of \(f(x)\)

Approximate the area under the graph of \(f(x)\) and above the \(x\) -axis, using each of the following methods with \(n=4\). (a) Use left endpoints. (b) Use right endpoints. (c) Average the answers in parts ( \(a\) ) and ( \(b\) ). (d) Use midpoints. $$f(x)=x+5 \text { from } x=2 \text { to } x=4$$

Let \(\lim _{x \rightarrow 4} f(x)=16\) and \(\lim _{x \rightarrow 4} g(x)=8 .\) Use the limit rules to find each limit. Do not use a calculator. $$\lim _{x \rightarrow 4}\left[\log _{2} f(x)\right]$$

Tell whether each statement is true or false. If \(\lim _{x \rightarrow 1} f(x)=5,\) then 1 must be in the domain of \(f(x)\)

Approximate the area under the graph of \(f(x)\) and above the \(x\) -axis, using each of the following methods with \(n=4\). (a) Use left endpoints. (b) Use right endpoints. (c) Average the answers in parts ( \(a\) ) and ( \(b\) ). (d) Use midpoints. $$f(x)=x+2 \text { from } x=0 \text { to } x=4$$

Find the slope of the tangent line to each curve when \(x\) has the given value. Do not use a calculator. $$f(x)=x^{2} ; x=4$$

Let \(\lim _{x \rightarrow 4} f(x)=16\) and \(\lim _{x \rightarrow 4} g(x)=8 .\) Use the limit rules to find each limit. Do not use a calculator. $$\lim _{x \rightarrow 4} \sqrt{f(x)}$$

Approximate the area under the graph of \(f(x)\) and above the \(x\) -axis, using each of the following methods with \(n=4\). (a) Use left endpoints. (b) Use right endpoints. (c) Average the answers in parts ( \(a\) ) and ( \(b\) ). (d) Use midpoints. $$f(x)=3+x \text { from } x=1 \text { to } x=3$$

Find the slope of the tangent line to each curve when \(x\) has the given value. Do not use a calculator. $$f(x)=x^{2}+5 ; x=2$$

Determine each limit. Refer to the accompanying graph of \(y=f(x)\) when it is given. Do not use a calculator. (a) \(\lim _{x \rightarrow 1^{+}} f(x)\) (b) \(\lim _{x \rightarrow 1^{-}} f(x)\) where \(f(x)=\frac{2 x}{(x-1)^{2}}\) (GRAPH CANNOT COPY).

Let \(\lim _{x \rightarrow 4} f(x)=16\) and \(\lim _{x \rightarrow 4} g(x)=8 .\) Use the limit rules to find each limit. Do not use a calculator. $$\lim _{x \rightarrow 4}[1+f(x)]^{2}$$

Tell whether each statement is true or false. If \(\lim _{x \rightarrow a} f(x)=b,\) then \(|f(x)-b|<0.0001\) for some value of \(x\) near \(a\)

Approximate the area under the graph of \(f(x)\) and above the \(x\) -axis, using each of the following methods with \(n=4\). (a) Use left endpoints. (b) Use right endpoints. (c) Average the answers in parts ( \(a\) ) and ( \(b\) ). (d) Use midpoints. $$f(x)=x^{2} \text { from } x=1 \text { to } x=5$$

Find the slope of the tangent line to each curve when \(x\) has the given value. Do not use a calculator. $$f(x)=-4 x^{2}+11 x ; x=-2$$

Let \(\lim _{x \rightarrow 4} f(x)=16\) and \(\lim _{x \rightarrow 4} g(x)=8 .\) Use the limit rules to find each limit. Do not use a calculator. $$\lim _{x \rightarrow 4} 2^{g(x)}$$

Determine each limit. Refer to the accompanying graph of \(y=f(x)\) when it is given. Do not use a calculator. $$\lim _{x \rightarrow 5^{+}}(3 x-5)$$

Approximate the area under the graph of \(f(x)\) and above the \(x\) -axis, using each of the following methods with \(n=4\). (a) Use left endpoints. (b) Use right endpoints. (c) Average the answers in parts ( \(a\) ) and ( \(b\) ). (d) Use midpoints. $$f(x)=-x^{2}+4 \text { from } x=-2 \text { to } x=2$$

Find the slope of the tangent line to each curve when \(x\) has the given value. Do not use a calculator. $$f(x)=6 x^{2}-4 x, x=-1$$

Let \(\lim _{x \rightarrow 4} f(x)=16\) and \(\lim _{x \rightarrow 4} g(x)=8 .\) Use the limit rules to find each limit. Do not use a calculator. $$\lim _{x \rightarrow 4} \sqrt[3]{g(x)}$$

Determine each limit. Refer to the accompanying graph of \(y=f(x)\) when it is given. Do not use a calculator. $$\lim _{x \rightarrow-4^{-}} x^{3}$$

Approximate the area under the graph of \(f(x)\) and above the \(x\) -axis, using each of the following methods with \(n=4\). (a) Use left endpoints. (b) Use right endpoints. (c) Average the answers in parts ( \(a\) ) and ( \(b\) ). (d) Use midpoints. $$f(x)=e^{x}-1 \text { from } x=0 \text { to } x=4$$

Find the slope of the tangent line to each curve when \(x\) has the given value. Do not use a calculator. $$f(x)=-\frac{2}{x} ; x=4$$

Let \(\lim _{x \rightarrow 4} f(x)=16\) and \(\lim _{x \rightarrow 4} g(x)=8 .\) Use the limit rules to find each limit. Do not use a calculator. $$\lim _{x \rightarrow 4} \frac{f(x)+g(x)}{2 g(x)}$$

Determine each limit. Refer to the accompanying graph of \(y=f(x)\) when it is given. Do not use a calculator. $$\lim _{x \rightarrow 7^{-}} 100$$

Approximate the area under the graph of \(f(x)\) and above the \(x\) -axis, using each of the following methods with \(n=4\). (a) Use left endpoints. (b) Use right endpoints. (c) Average the answers in parts ( \(a\) ) and ( \(b\) ). (d) Use midpoints. $$f(x)=e^{x}+1 \text { from } x=-2 \text { to } x=2$$

Find the slope of the tangent line to each curve when \(x\) has the given value. Do not use a calculator. $$f(x)=\frac{6}{x} ; x=-1$$

Let \(\lim _{x \rightarrow 4} f(x)=16\) and \(\lim _{x \rightarrow 4} g(x)=8 .\) Use the limit rules to find each limit. Do not use a calculator. $$\lim _{x \rightarrow 4} \frac{5 g(x)+2}{1-f(x)}$$

Approximate the area under the graph of \(f(x)\) and above the \(x\) -axis, using each of the following methods with \(n=4\). (a) Use left endpoints. (b) Use right endpoints. (c) Average the answers in parts ( \(a\) ) and ( \(b\) ). (d) Use midpoints. $$f(x)=\frac{1}{x} \text { from } x=1 \text { to } x=5$$

Find the slope of the tangent line to each curve when \(x\) has the given value. Do not use a calculator. $$f(x)=-3 \sqrt{x} ; x=1$$

Determine each limit, if it exists. $$\lim _{x \rightarrow-3} 7$$

Determine each limit. Refer to the accompanying graph of \(y=f(x)\) when it is given. Do not use a calculator. $$\lim _{x \rightarrow-3^{-}} \sqrt{x+3}$$

Approximate the area under the graph of \(f(x)\) and above the \(x\) -axis, using each of the following methods with \(n=4\). (a) Use left endpoints. (b) Use right endpoints. (c) Average the answers in parts ( \(a\) ) and ( \(b\) ). (d) Use midpoints. $$f(x)=\frac{2}{x} \text { from } x=1 \text { to } x=9$$

Find the slope of the tangent line to each curve when \(x\) has the given value. Do not use a calculator. $$f(x)=x^{3} ; x=1$$

Determine each limit, if it exists. $$\lim _{x \rightarrow 6}(-5)$$

Determine each limit. Refer to the accompanying graph of \(y=f(x)\) when it is given. Do not use a calculator. $$\lim _{x \rightarrow 2^{-}} \sqrt{2-x}$$

Consider the region below \(f(x)=\frac{x}{2},\) above the \(x\) -axis, and between \(x=0\) and \(x=4 .\) Let \(x_{i}\) be the midpoint of the \(i\) th subinterval. (a) Approximate the area of the region, using four rectangles. (b) Find \(\int_{0}^{4} f(x) d x\) by using the formula for the area of a triangle.

Find the slope of the tangent line to each curve when \(x\) has the given value. Do not use a calculator. $$f(x)=2 x^{3} ; x=1$$

Determine each limit, if it exists. $$\lim _{x \rightarrow \pi} x$$

Determine each limit. Refer to the accompanying graph of \(y=f(x)\) when it is given. Do not use a calculator. $$\lim _{x \rightarrow 0^{-}} \frac{|x|}{x}$$

Find \(\int_{0}^{5}(5-x) d x\) by using the formula for the area of a triangle.