Problem 1
A particle starts at \(x=0\) and moves along the \(x\) -axis with velocity \(v(t)=5\) for time \(t \geq 0\) . Where is the particle at \(t=4\) ?
Problem 1
In Exercises \(1-20,\) find \(d y / d x\). $$y=\int_{0}^{x}\left(\sin ^{2} t\right) d t$$
Problem 1
In Exercises 1-6, (a) use the Trapezoidal Rule with n = 4 to approximate the value of the integral. (b) Use the concavity of the function to predict whether the approximation is an overestimate or an underestimate. Finally, (c) find the integral's exact value to check your answer. $$\int_{0}^{2} x d x$$
Problem 1
In Exercises \(1-6,\) each \(c_{k}\) is chosen from the \(k\) th subinterval of a regular partition of the indicated interval into \(n\) subintervals of length \(\Delta x .\) Express the limit as a definite integral. $$\lim _{n \rightarrow \infty} \sum_{k=1}^{n} c_{k}^{2} \Delta x,[0,2]$$
Problem 2
In Exercises \(1-6,\) each \(c_{k}\) is chosen from the \(k\) th subinterval of a regular partition of the indicated interval into \(n\) subintervals of length \(\Delta x .\) Express the limit as a definite integral. $$\lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left(c_{k}^{2}-3 c_{k}\right) \Delta x,[-7,5]$$
Problem 2
Suppose that \(f\) and \(h\) are continuous functions and that \(\int _ { 1 } ^ { 9 } f ( x ) d x = - 1 , \quad \int _ { 7 } ^ { 9 } f ( x ) d x = 5 , \quad \int _ { 7 } ^ { 9 } h ( x ) d x = 4\) Use the rules in Table 5.3 to find each integral. (a) $$\int _ { 1 } ^ { 9 } - 2 f ( x ) d x \quad \quad$$ (b) $$\int _ { 7 } ^ { 9 } [ f ( x ) + h ( x ) ] d x$$ $$( \mathbf { c } ) \int _ { 7 } ^ { 9 } [ 2 f ( x ) - 3 h ( x ) ] d x \quad$$ (d) $$\int _ { 9 } ^ { 1 } f ( x ) d x$$ (e) $$\int _ { 1 } ^ { 7 } f ( x ) d x \quad$$ (f) $$\int _ { 9 } ^ { 7 } [ h ( x ) - f ( x ) ] d x$$
Problem 2
A particle starts at \(x=0\) and moves along the \(x\) -axis with velocity \(v(t)=2 t+1\) for time \(t \geq 0 .\) Where is the particle at \(t=4 ?\)
Problem 2
In Exercises 1-6, (a) use the Trapezoidal Rule with n = 4 to approximate the value of the integral. (b) Use the concavity of the function to predict whether the approximation is an overestimate or an underestimate. Finally, (c) find the integral's exact value to check your answer. $$\int_{0}^{2} x^{2} d x$$
Problem 2
In Exercises \(1-20,\) find \(d y / d x\). $$y=\int_{2}^{x}\left(3 t+\cos t^{2}\right) d t$$
Problem 3
A particle starts at \(x=0\) and moves along the \(x\) -axis with velocity \(v(t)=t^{2}+1\) for time \(t \geq 0 .\) Where is the particle at \(t=4 ?\) Approximate the area under the curve using four rectangles of equal width and heights determined by the midpoints of the intervals, as in Example \(1 .\)
