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From the first principles, prove that the derivative of the functionf(x

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MA1S11 (Dotsenko) Sample questions for the calculus part of 1S11

Michaelmas 2013 1. Compute the limit lim

x→0 tan(7x)

e3x−1.

2. From the first principles, prove that the derivative of the functionf(x) = 1x is given by the formula 2x1x.

3. Is the function

f(x) =

(x3sin1x, x6= 0, 0, x= 0

continuous at x= 0? differentiable atx= 0? twice differentiable at x= 0? Explain your answer.

4. Compute the derivatives:

(a) (tan(7 + 5 lnx))3; (b) cos−1x; (c) x1/x; (d) ln

ex 1+ex

. 5. Compute f(π/6), if f(x) = tan1(cosx).

6. Compute f(e) for f(x) = lnx3x.

7. “The slope of the tangent to the curvey =ax3+bx+ 4 at the point (2,14) on that curve is 21.” Find the values ofaand b for which it is true.

8. Forf(x) = sin(lnx), show that x2f′′+xf+f = 0.

9. Determine relative extrema and inflection points of the graph y = x3 −8x2 + 16x, and draw a rough sketch of that graph.

10. Show that among all the rectangles of area A, the square has the minimum perimeter.

11. The concentration C of an antibiotic in the bloodstream after time tis given by C = 5t

1 +kt22

for a certain constantk. If it is known that the maximal concentration is reached att= 6 hours, find the value ofk.

12. Evaluate the integrals (a)R sin 2θ

1+cos 2θdθ; (b)R x dx

1+x2; (c)R x33

1−4x dx.

13. Evaluate the integrals (a)

R1 1/2

3

2xdx; (b) Rπ 0

cos2x

1+sinxdx; (c) Re e1

1−(lnx)2

x dx.

14. Find a positive value ofk for which the area under the graph ofy=e3x over the interval [0, k] is 11 square units.

15. Compute the area of the region between the graphs y=xex and y=x2ex.

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